CRC Handbook of Chemistry and Physics

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CRC Handbook of Chemistry and Physics Editor-in-Chief David R. Lide Former Director, Standard Reference Data National Institute of Standards and Technology

Editorial Advisory Board Grace Baysinger Swain Chemistry and Chemical Engineering Library Stanford University Stanford, CA 94305-5080 Lev I. Berger California Institute of Electronics and Materials Science 2115 Flame Tree Way Hemet, CA 92545 Robert N. Goldberg Biotechnology Division National Institute of Standards and Technology Gaithersburg, MD 20899 Henry V. Kehiaian ITODYS University of Paris VII 1, rue Guy de la Brosse 75005 Paris, France Kozo Kuchitsu Department of Chemistry Josai University, Sakado 350-0295, Japan Gerd Rosenblatt 1177 Miller Avenue Berkeley, CA 94708 Dana L. Roth Millikan Library / Caltech 1-32 1200 E. California Blvd. Pasadena, CA 91125 Daniel Zwillinger Mathematics Department Rensselaer Polytechnic Institute Troy, NY 12180

FOREWORD My acquaintance with the CRC Handbook goes back sixty years, for when I was inducted into the wonders of chemistry by an uncle of mine (“Uncle Tungsten”)—I was ten—he lent me his copy of the 23rd (1939) edition. This was not pocket-sized, like the earlier editions he had on his shelf, and indeed contained over 2200 pages, but these were printed on thin India paper, and the whole book, with its soft red morocco cover, fitted easily in the hand. I fell in love with it straightaway—my uncle, seeing this, told me I might keep it—for its tables were so full of information that I thought of it as containing the whole universe between its covers. I was especially attracted to the Physical Constants of Inorganic Compounds, a hundred and fifty densely-packed pages which, through constant poring over, I got almost by heart. I think I owe the only original idea I had in my chemical boyhood to these tables—for, having been struck by the steadily rising melting points and densities of the transition metals in Groups IV-VI as one went from Period 3 to 6 (Ti, Zr, Hf; V, Nb, Ta; Cr, Mo, W), I was then taken aback to find that the Period 7 analogues of these broke the series. Thorium had a lower melting point and density than hafnium; uranium lower ones than tungsten. Could it be, I wondered, that they were not in fact analogues of hafnium and tungsten, not transition metals at all, but belonged to an interpolated series which resembled the rare-earth metals? To my joy, after the War, I found that this naïf idea of mine, a possibly unjustified leap of the imagination, turned out to be true—but it was entirely due to poring over the tables of the CRC Handbook that I owed it. Although my interests later turned more to biology and then medicine, the CRC Handbook has never lost its enchantment for me. I got the 30th (1947) and the 41st (1959-1960) editions—at this point the Handbook still had its smaller format, but had become almost cubical in shape (the 41st edition had nearly 3500 pages); and then, of course, it morphed into its present, monumental format. While I keep the massive recent editions in my study, I keep my original one, the 23rd edition, on my bedside table, for it is easy to handle (especially when one is reading in bed), and was my most cherished gift as a boy. Indeed, one way and another, whether reading in bed or in my study, I have always had a Handbook near me. While the CRC Handbook is monumental in its scope, a huge, alwaysto-be-relied-upon mine of information, it is also a friendly book, a companion which has given me joy for the greater part of my life. Oliver Sacks New York October 2003

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PREFACE Since the First Edition of the CRC Handbook of Chemistry and Physics appeared in 1913, the size and scope have expanded in step with the growth of scientific knowledge. It has not only served as a reference source for professionals and students, but has provided inspiration to many young people as they developed their interest in science. The late Linus Pauling, in his Foreword to the 74th Edition, wrote "I attribute much of my knowledge about substances and their properties to my study of the information that the Handbook provided." In the Foreword to the present edition Oliver Sacks, author of the best seller Uncle Tungsten: Memories of a Chemical Boyhood, describes the strong influence the Handbook had on him from the age of ten. Throughout its history the overall philosophy of the Handbook has been to provide broad coverage of all types of data commonly encountered by physical scientists and engineers. While the Internet has spawned numerous large databases covering narrow areas of science, we feel there is still a need for a concise reference source spanning the full range of the physical sciences and focusing on key data that are frequently needed by R&D professionals, engineers, and students. We hope this Internet version of the CRC Handbook will be a step in continuing to serve these needs. The 85th Edition includes updates and expansions of several tables, such as Aqueous Solubility of Organic Compounds, Thermal Conductivity of Liquids, and Table of the Isotopes. A new table on Azeotropic Data for Binary Mixtures has been added, as well as tables on Index of Refraction of Inorganic Crystals and Critical Solution Temperatures of Polymer Solutions. In response to user requests, several topics such as Coefficient of Friction and Miscibility of Organic Solvents have been restored to the Handbook. The latest recommended values of the Fundamental Physical Constants, released in December 2003, are included in this edition. Finally, the Appendix on Mathematical Tables has been revised by Dr. Daniel Zwillinger, editor of the CRC Standard Mathematical Tables and Formulae; it includes new information on factorials, Clebsch-Gordan coefficients, orthogonal polynomials, statistical formulas, and other topics. This new Internet edition has added 13 new subsections that can be accessed as interactive tables. These include tables on atomic and molecular polarizabilities, diffusion in gases and liquids, vapor pressure and density of mercury, ionic radii in crystals, surface tension, and other topics. All material in the printed Handbook is accessible in the Internet version as interactive tables and/or pdf displays. The Editor appreciates suggestions on new topics for the Handbook and notification of any errors. Input from users plays a key role in keeping the book up to date. Address all comments to Editor-in-Chief, Handbook of Chemistry and Physics, CRC Press LLC, 2000 N. W. Corporate Blvd., Boca Raton, FL 33431. Comments may also be sent by electronic mail to [email protected].

The Handbook of Chemistry and Physics is dependent on the efforts of many contributors throughout the world. Valuable suggestions have been received from the Editorial Advisory Board and from many users. The assistance and support of Dr. Fiona Macdonald, Chemistry Publisher at CRC Press, is greatly appreciated. Finally, I want to thank Susan Fox, James Miller, Helena Redshaw, James Yanchak, Robert Morris, and Ronel Decius of the CRC Press staff for all their efforts. David R. Lide October 2004

How To Cite this Reference The recommended form of citation is: David R. Lide, ed., CRC Handbook of Chemistry and Physics, Internet Version 2005, , CRC Press, Boca Raton, FL, 2005. If a specific table is cited, use the format: "Physical Constants of Organic Compounds", in CRC Handbook of Chemistry and Physics, Internet Version 2005, David R. Lide, ed., , CRC Press, Boca Raton, FL, 2005.

This work contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Best efforts have been made to select and verify the data on the basis of sound scientific judgment, but the author and the publisher cannot accept responsibility for the validity of all materials or for the consequences of their use. © Copyright CRC Press LLC 2005

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CURRENT CONTRIBUTORS Lev I. Berger California Institute of Electronics and Materials Science 2115 Flame Tree Way Hemet, California 92545 A. K. Covington Department of Chemistry University of Newcastle Newcastle upon Tyne NE1 7RU England K. Fischer LTP GmbH Oppelner Strasse 12 D-26135 Oldenburg, Germany Jean-Claude Fontaine ITODYS CNRS, University of Paris VII 1 rue Guy de la Brosse 75005 Paris, France H. P. R. Frederikse 9625 Dewmar Lane Kensington, Maryland 20895 J.R. Fuhr Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 J. Gmehling Universität Oldenburg Fakultät V, Technische Chemie D-26111 Oldenburg, Germany Robert N. Goldberg Biotechnology Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 C. R. Hammond 17 Greystone Rd. West Hartford, Connecticut 06107

Norman E. Holden National Nuclear Data Center Brookhaven National Laboratory Upton, New York 11973 H. Donald Brooke Jenkins Department of Chemistry University of Warwick Coventry CV4 7AL England Henry V. Kehiaian ITODYS University of Paris VII 1 rue Guy de la Brosse 75005 Paris, France J. Alistair Kerr School of Chemistry University of Birmingham Birmingham B15 2TT England J. Krafczyk DDBST GmbH Industriestrasse 1 D-26121 Oldenburg, Germany Frank J. Lovas 8616 Melwood Rd. Bethesda, Maryland 20817 William C. Martin Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 J. Menke DDBST GmbH Industriestrasse 1 D-26121 Oldenburg, Germany Thomas M. Miller Air Force Research Laboratory/VSBP 29 Randolph Rd. Hanscom AFB, Massachusetts 01731-3010

Peter J. Mohr Physics Laboratory National Institute of Standards and Technology Gaithersburg, Maryland 20899 Joseph Reader Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 Lewis E. Snyder Astronomy Department University of Illinois Urbana, Illinois 61801 B. N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, Maryland 20899 Thomas G. Trippe Particle Data Group Lawrence Berkeley Laboratory 1 Cyclotron Road Berkeley, California 94720 Petr Vany´sek Department of Chemistry Northern Illinois University DeKalb, Illinois 60115 Wolfgang L. Wiese Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 Christian Wohlfarth Institut für Physikalische Chemie Martin Luther University D-06217 Merseburg Germany Daniel Zwillinger Mathematics Department Rensselaer Polytechnic Institute Troy, New York 12180

Section 1: Basic Constants, Units, and Conversion Factors Fundamental Physical Constants Standard Atomic Weights (2001) Atomic Masses and Abundances Electron Configuration of Neutral Atoms in the Ground State International Temperature Scale of 1990 (ITS-90) Conversion of Temperatures from the 1948 and 1968 Scales to ITS-90 International System of Units (SI) Units for Magnetic Properties Conversion Factors Conversion of Temperatures Conversion Factors for Energy Units Conversion Factors for Pressure Units Conversion Factors for Thermal Conductivity Units Conversion Factors for Electrical Resistivity Units Conversion Factors for Chemical Kinetics Conversion Factors for Ionizing Radiation Values of the Gas Constant in Different Unit Systems Periodic Table of the Elements

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)81'$0(17$/3+  +π 20-8 
 #2π
 

  ?
 + & :  + &'' (4  + & :  + &'&12% (4  0) x 2 b 0 b b b The area enclosed by a curve defined through the equation x c + y c = a c where a > 0, c  2  c [Γ( )] 2ca2 a positive odd integer and b a positive even integer is given by Γ b2c b (b) 0

I =

xh−1 y m−1 z n−1 dv, where R denotes the region of space bounded by the co-

R  p  q  k ordinate planes and that portion of the surface xa + yb + zc = 1, which lies in the first octant and where h, m, n, p, q, k, a, b, c, denote positive real numbers is given by a h c  x p  1  x p  y q  1 e m e n−1 h−1 x dx y dy − z dz 1− 1− a a b 0 0      0 h m n ah bm cn Γ p Γ q Γ k   = pqk Γ h + m + n + 1 p q k ∞ 1 −ax e dx = , (a > 0) a 0 ∞ −ax b e − e−bx dx = log , (a, b > 0) x a 0  Γ(n+1)  n > −1, a > 0 ∞  an+1 xn e−ax dx = or  0  n! (a > 0, n positive integer) an+1   ∞ Γ(k) n+1 n p x exp(−ax )dx = , n > −1, p > 0, a > 0, k = pak   p 0 ∞ √ 2 2 1 1 1 −a x e dx = π= Γ , (a > 0) 2a 2a 2 0 ∞ 2 1 xe−x dx = 2 0 √ ∞ 2 π x2 e−x dx = 4 0

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INTEGRALS (Continued)

1 · 3 · 5 . . . (2n − 1) π (a > 0, n > − 12 ) 2n+1 an a 0 ∞ 2 n! x2n+1 e−ax dx = n+1 , (a > 0, n > −1) 2a

0  1 m  m! ar m −ax −a x e dx = m+1 1 − e a r! 0 r=0 √ ∞  2 a2  −2a −x − 2 e π x e dx = , (a ≥ 0) 2

0 ∞ √ 1 π e−nx x dx = (n > 0) 2n n 0

∞ −nx e π √ dx = (n > 0) n x 0 ∞ a e−ax (cos mx) dx = 2 , (a > 0) a + m2 0 ∞ m e−ax (sin mx) dx = 2 , (a > 0) a + m2 0 ∞ 2ab xe−ax [sin(bx)] dx = 2 , (a > 0) (a + b2 )2 0 ∞ a2 − b2 xe−ax [cos(bx)] dx = 2 , (a > 0) (a + b2 )2 0 ∞ n![(a + ib)n+1 − (a − ib)n+1 ] xn e−ax [sin(bx)] dx = , (i2 = −1, a > 0) 2i(a2 + b2 )n+1 0 ∞ n![(a − ib)n+1 + (a + ib)n+1 ] xn e−ax [cos(bx)] dx = , (i2 = −1, a > 0, n > −1) 2 + b2 )n+1 2(a 0 ∞ −ax e sin x dx = cot−1 a, (a > 0) x 0   √ ∞ b2 π −a2 x2 e cos bx dx = exp − 2 , (ab = 0) 2|a| 4a 0 ∞  π π −t cos φ b−1 e t [sin(t sin φ)] dt − [Γ(b)] sin(bφ), b > 0, − < φ < 2 2 0 ∞  π π e−t cos φ tb−1 [cos(t sin φ)] dt − [Γ(b)] cos(bφ), b > 0, − < φ < 2 2   0 ∞ bπ b−1 t cos t dt = [Γ(b)] cos , (0 < b < 1) 2  0 ∞ bπ tb−1 (sin t) dt = [Γ(b)] sin , (0 < b < 1) 2 0 1 (log x)n dx = (−1)n · n! (n > −1) 0 1 √ 1 1 2 π dx = log x 2 0 − 1 1 2 √ 1 dx = π log x 0 1  n 1 dx = n! log x 0 1 3 x log(1 − x) dx = − 4 0 1 1 x log(1 + x) dx = 4 0 1 (−1)n n! xm (log x)n dx = , m > −1, n = 0, 1, 2, . . . (m + 1)n+1 0 If n = 0, 1, 2, . . . replace n! by Γ(n + 1).

666. 667. 668. 669. 670. 671. 672. 673. 674. 675. 676. 677. 678. 679. 680. 681. 682. 683. 684. 685. 686. 687. 688. 689. 690.

∞

2

x2n e−ax dx =

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INTEGRALS (Continued)

1

691. 0 1 692. 0 1 693. 0 1 694. 0 1 695. 0 1 696. 0 1

697. 698. 699. 700. 701. 702. 703. 704. 705. 706. 707. 708. 709. 710. 711. 712. 713. 714. 715. 716.

π2 log x dx = − 1+x 12 π2 log x dx = − 1−x 6 π2 log(1 + x) dx = x 12 π2 log(1 − x) dx = − x 6 (log x)[log(1 + x)] dx = 2 − 2 log 2 − (log x)[log(1 − x)] dx = 2 −

π2 12

π2 6

π2 log x dx = − 2 1 −x  8 0 1 1+x π2 dx log = · 1 − x x 4 0 1 π log x dx √ = − log 2 2 2 0 1 1 − x   n Γ(n + 1) 1 m x log dx = , if m + 1 > 0, n + 1 > 0 x (m +1)n+1  0 1 p p+1 (x − xq )dx = log , (p + 1 > 0, q + 1 > 0) log x q+1 0 1 √ dx    = π, (same as integral 686) 0 log x1  x  ∞ π2 e +1 log dx = x e −1 4 0 π/2 π/2 π (log sin x) dx = log cos x dx = − log 2 2 0 0 π/2 π/2 π (log sec x) dx = log csc x dx = log 2 2 0 0 π π2 x(log sin x) dx = − log 2 2 0 π/2 (sin x)(log sin x) dx = log 2 − 1 0 π/2 (log tan x) dx = 0 √   0π a + a 2 − b2 log(a ± b cos x) dx = π log , (a ≥ b) 2 0 $ π 2π log a a≥b>0 log(a2 − 2ab cos x + b2 ) dx = 2π log b b≥a>0 0 ∞ π aπ sin ax dx = tanh 2|b| 2b 0 ∞ sinh bx π aπ cos ax dx = sech 2|b| 2b 0 ∞ cosh bx π dx = cosh ax 2|a| 0 ∞ π2 x dx = 2 (a > 0) 4a 0 ∞ sinh ax a −ax e (cosh bx) dx = 2 , (0 ≤ |b| < a) a − b2 0 ∞ b e−ax (sinh bx) dx = 2 , (0 ≤ |b| < a) a − b2 0

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INTEGRALS (Continued)

717. 718. 719.

∞

π aπ 1 sinh ax dx = csc − (b > 0) bx + 1 e 2b b 2a 0 ∞ 1 π aπ sinh ax dx = − cot (b > 0) ebx − 1 2a 2b b 0    2 2  2 π/2 π 1·3 1·3·5 dx 1 2 4 6
= k + k + k + ··· , 1+ 2 2 2·4 2·4·6 0 1 − k2 sin2 x if k2 < 1 

 2 4   6  2 π/2
π 1·3 1·3·5 k k 1 2 2 2 1 − k sin x dx = k − − − ··· , 2 1− 2 2 2·4 3 2·4·6 5 0

720.



721. 722. 723. 724.

725. 726.

727. 728.

if k2 < 1

∞

e−x log x dx = −γ = −0.5772157 . . . √ ∞ 2 π e−x log x dx = − (γ + 2 log 2) 4  0 ∞  1 1 − [Euler’s Constant] e−x dx = γ = 0.5772157 . . . −x 1 − e x 0  ∞  1 1 − e−x dx = γ = 0.5772157 . . . x 1 + x 0 For n even :     n/2−1  n sin(n − 2k)x 1 n 1 + n cosn x dx = n−1 x k 2 (n − 2k) 2 n/2 k=0     n/2−1  n sin[(n − 2k)( π −x)] 1 1 n n 2 + n x sin x dx = n−1 k 2 2k − n 2 n/2 k=0 For n odd:   (n−1)/2  n sin(n − 2k)x 1 n cos x dx = n−1 k 2 n − 2k k=0     π (n−1)/2  n sin (n − 2k) 2 −x 1 sinn x dx = n−1 k 2 2k–n

0

k=0

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2k − n

DIFFERENTIAL EQUATIONS

SPECIAL FORMULAS Certain types of differential equations occur sufficiently often to justify the use of formulas for the corresponding particular solutions. The following set of tables I to XIV covers all first, second, and nth order ordinary linear differential equations with constant coefficients for which the right members are of the form P (x)erx sin sx or P (x)erx cos sx, where r and s are constants and P (x), is a polynomial of degree n. When the right member of a reducible linear partial differential equation with constant coefficients is not zero, particular solutions for certain types of right members are contained in tables XV to XXI. In these tables both F and P are used to denote polynomials, and it is assumed that no denominator is zero. In any formula the roles of x and y may be reversed throughout, changing a formula in which x dominates to one in which y dominates. XXI are applicable   Tables XIX, XX, m! st whether the equations are reducible or not. The symbol m stands for n (m−n)!n! and is the n + 1 coefficient in the expansion of (a + b)m . Also 0! = 1 by definition.

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DIFFERENTIAL EQUATIONS (Continued) Solution of Linear Differential Equations with Constant Coefficients Any linear differential equation with constant coefficients may be written in the form p(D)y = R(x) where • • • • •

D is the differential operation: Dy = p(D) is a polynomial in D, y is the dependent variable, x is the independent variable, R(x) is an arbitrary function of x.

dy dx

A power of D represents repeated differentiation, that is dn y dxn For such an equation, the general solution may be written in the form Dn y =

y = yc + yp where yp is any particular solution, and yc is called the complementary function. This complementary function is defined as the general solution of the homogeneous equation, which is the original differential equation with the right side replaced by zero, i.e. p(D)y = 0 The complementary function yc may be determined as follows: 1. Factor the polynomial p(D) into real and complex linear factors, just as if D were a variable instead of an operator. 2. For each nonrepeated linear factor of the form (D - a), where a is real, write down a term of the form ceax where c is an arbitrary constant. 3. For each repeated real linear factor of the form (D − a)n , write down n terms of the form c1 eax + c2 xeax + c3 x2 eax + · · · + cn xn−1 eax where the ci ’s are arbitrary constants. 4. For each non-repeated conjugate complex pair of factors of the form (D − a + ib)(D − a − ib), write down 2 terms of the form c1 eax cos bx + c2 eax sin bx 5. For each repeated conjugate complex pair of factors of the form (D − a + ib)n (D − a − ib)n , write down 2n terms of the form c1 eax cos bx + c2 eax sin bx + c3 xeax cos bx + c4 xeax sin bx + · · · + c2n−1 xn−1 eax cos bx + c2n xn−1 eax sin bx 6. The sum of all the terms thus written down is the complementary function yc . To find the particular solution yp , use the following tables, as shown in the examples. For cases not shown in the tables, there are various methods of finding yp . The most general method is called variation of parameters. The following example illustrates the method:

A-57

DIFFERENTIAL EQUATIONS (Continued) Example: Find yp for (D2 − 4)y = ex . This example can be solved most easily by use of equation 63 in the tables following. However it is given here as an example of the method of variation of parameters. The complementary function is yc = c1 e2x + c2 e−2x To find yp , replace the constants in the complementary function with unknown functions, yp = ue2x + ve−2x We now prepare to substitute this assumed solution into the original equation. We begin by taking all the necessary derivatives: yp = ue2x + ve−2x yp = 2ue2x − 2ve−2x + u e2x + v  e−2x For each derivative of yp except the highest, we set the sum of all the terms containing u and v to 0. Thus the above equation becomes u e2x + v  e−2x = 0

and yp = 2ue2x − 2ve−2x

Continuing to differentiate, we have yp = 4ue2x + 4ve−2x + 2u e2x − 2v  e−2x When we substitute into the original equation, all the terms not containing u or v cancel out. This is a consequence of the method by which yp was set up. Thus all that is necessary is to write down the terms containing u or v in the highest order derivative of yp , multiply by the constant coefficient of the highest power of D in p(D), and set it equal to R(x). Together with the previous terms in u and v which were set equal to 0, this gives us as many linear equations in the first derivatives of the unknown functions as there are unknown functions. The first derivatives may then be solved for by algebra, and the unknown functions found by integration. In the present example, this becomes u e2x + v  e−2x = 0 2u e2x − 2v  e−2x = ex We eliminate v and u separately, getting 4u e2x = ex 4v  e−2x = −ex Thus

Therefore, by integrating

u = 14 e−x v  = − 14 e3x u = − 14 e−x 1 3x v = − 12 e

A constant of integration is not needed, since we need only one particular solution. Thus 1 1 yp = ue2x + ve−2x = − e−x e2x − e3x e−2x 4 12 1 1 1 = − ex − ex = − ex 4 12 13

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DIFFERENTIAL EQUATIONS (Continued) and the general solution is y = yc + yp = c1 e2x + c2 e−2x −

1 x e 3

The following samples illustrate the use of the tables. Example 1:

Solve (D2 − 4)y = sin 3x. Substitution of q = −4, s = 3 in formula 24 gives yp =

sin 3x −9 − 4

wherefore the general solution is y = c1 e2x + c2 e−2x −

sin 3x 13

Example 2: Obtain a particular solution of (D2 − 4D + 5)y = x2 e3x sin x. Applying formula 40 with a = 2, b = 1, r = 3, s = 1, P (x) = x2 , s + b = 2, s − b = 0, a − r = −1, (a − r)2 +(s + b)2 =5, (a − r)2 +(s − b)2 = 1, we have

yp

=

=

     2(−1)2 0 2(−1)0 3·1·0−0 2 3 · 1 · 2 − 23 2 − − − x + 2x + 2 5 1 25 1 125 1       1−4 −1 1−0 −1 − 3(−1)0 −1 −1 − 3(−1)4 e3x cos x − − − x2 + 2x + 2 − 2 5 1 25 1 125 1     1 2 4 2 28 2 136 x − x− − e3x sin x + − x2 + e3x cos x 5 25 125 5 25 125 e3x sin x 2



The special formulas effect a very considerable saving of time in problems of this type. Example 3: Obtain a particular solution of (D2 − 4D + 5)y = x2 e2x cos x. (Compare with Example 2.) Formula 40 is not applicable here since for this equation r = a, s = b, wherefore the denominator (a − r)2 +(s − b)2 = 0. We turn instead to formula 44. Substituting a = 2, b = 1, P (x) = x2 and replacing sin by cos, cos by -sin, we obtain e2x cos x 2 2 e2x sin x yp = (x −4 ) + (x2 −12 ) dx 4 2  3    2 x 1 x x − − e2x cos x + e2x sin x = 4 8 6 4 which is the required solution. Example 4: Find zp for (Dx − 3Dy ) z = ln(y + 3x). Referring to Table XV we note that formula 69 (not 68) is applicable. This gives zp = x ln(y + 3x) It is easily seen that −y/3 ln(y + 3x) would serve equally well. Example 5: Solve (Dx + 2Dy − 4) z = y cos(y − 2x). Since R in formula 76 contains a polynomial in x, not y, we rewrite the given equation in the form (Dy +12 Dx − 2) z =12 y cos(y − 2x). Then zc = e2y F (x − 1 y) = e2x f (2x − y) 2 and by the formula 1 zp = − cos(y − 2x) · 2



1 y + 2 2 2



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1 = − (2y + 1) cos(y − 2x) 8

DIFFERENTIAL EQUATIONS (Continued) Example 6: Find zp for (Dx + 4Dy )3 z = (2x − y)2 . Using formula 79, we obtain  2 3 u du u5 (2x − y)5 zp = = =− 3 [2 + 4(−1)] 5 · 4 · 3 · (−8) 480 Example 7:

Find zp for (Dx3 + 5Dx2 Dy − 7Dx + 4)z = e2x+3y . By formula 87 zp =

Example 8:

e2x+3y e2x+3y = 23 + 5 · 22 · 3 − 7 · 2 + 4 58

Find zp for (Dx4 + 6Dx3 Dy + Dx Dy + Dy2 + 9)z = sin(3x + 4y)

Since every term in the left number is of even degree in the two operators Dx and Dy , formula 90 is applicable. It gives sin(3x + 4y) (−9)2 + 6(−9)(−12) + (−12) + (−16) + 9 sin(3x + 4y) = 710

zp =

Table I: (D − a)y = R R 1. erx 2. sin sx∗ 3. P (x) 4. erx sin sx∗ 5. P (x) erx 6. P (x) sin sx∗

yp

erx r−a cos sx − a sin sx+s a2 +s2

 − a1 P (x) +

= √

P  (x) a

+

1 a2 +s2

P  (x) a2

  sin sx + tan−1 as (n) + · · · + P an(x)

Replace a by a − r in formula 2 and multiply by erx . rx Replace a by  a − r in formula 3 and multiply by e .

a2 −s2 a3 −3as2 a   2 +s2 P (x) + (a2 +s2 )2 P (x) + (a2 +s2 )3 P (x) a     k k k k−2 2 k−4 4 a − 2 a s + 4 a s −··· P (k−1) (x) + · · · (a2 +s2 )k

+ ···

3a2 s−s3 s 2as   2 2 P (x) + (a2 +s2 )2 P (x) + (a2 +s2 )3 P (x)     a +s k k ak−1 s− 3 ak−3 s3 +··· + 1 P (k−1) (x) + · · · (a2 +s2 )k

+ ···

− sin sx +

− cos sx



7. P (x)erx sin sx∗ Replace a by a − r in formula 6 and multiply by erx . 8. eax xeax ax sx 9. eax sin sx∗ − e  cos s 10. P (x)eax eax P (x) dx  11. P (x)eax sin sx

  v ax sx P (x) − − P s3(x) + P s5(x) − · · · − e cos s *For cos sx in R replace “sin” by “cos” and “cos” by “−sin” in yp . eax sin sx s

P  (x) s3

Dn =

dn dxn

(m n )=

m! (m − n)!n!

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0! = 1

P  (x) s2

+

P iv (x) s4

− ···



DIFFERENTIAL EQUATIONS (Continued) Table II: (D − a)2 y = R R 12. erx

yp

erz (r−a)2 1 [(a2 (a2  +s2 )

13. sin sx∗

− s2 ) sin sx + 2as cos sx] = 2P  (x) a

3P  (x) a2

1 a2 +s2

 sin sx + tan−1

(n+1)P (n) (x) an multiply by erx .

1 a2

15. erx sin sx∗ 16. P (x)erx

Replace a by a − r in formula 13 and Replace a by a − r in formula 14 and multiply by erx .

17. P (x) sin sx∗

+

a2 −s2 a3 −3as2 a4 −6a2 s2 +s4   P (x) + 2 (a P (x) 2 +s2 )3 P (x) + 3 (a2 +s2 )2 (a2 +s2 )4      ak − k ak−2 s2 + k ak−4 s4 −··· 2 4 (k−2) (x) + · · · +(k − 1) P (a2 +s2 )k

sin sx



+ cos sx

2as P (x) + (a2 +s2 )2  

+(k − 1)

k 1

2

3

3

ak−1 s− k ak−3 s3 +··· 3 (a2 +s2 )k

3

P (k−2) (x) + · · ·

 iv vi sin sx P (x) − 3Ps2(x) + 5P s4(x) − 7P s6(x) s2     v ax sx 2P (x) − e scos + 4P s3(x) − 6Ps5(x) − · · · 2 s

−e

+ ···

3a s−s 4a s−4as   2 (a 2 +s2 )3 P (x) + 3 (a2 +s2 )4 P (x) + · · ·   

18. P (x)erx sin sx∗ Replace a by a − r in formula 17 and multiply by erx . 1 2 ax 19. eax x e 2 ax sx 20. eax sin sx∗ − e  ssin 2 P (x) dx 21. P (x)eax eax  dx 22. P (x)eax sin sx∗



+ ··· +

14. P (x)

P (x) +

2as a2 −s2

ax

+ ···



* For cos sx in R replace “sin” by “cos” and “cos” by “-sin” in yp .

Table III: (D 2 + q)y = R R 23. erx 24. sin sx∗ 25. P (x) 26. erx sin sx 27. P (x)erx

28. P (x) sin sx∗

yp

erx r 2 +q sin sx −s2 +q   iv (2k) 1 P (x) − P q(x) + P q2(x) − · · · + (−1)k P qk (x) · · · q  (r 2 −s2 +q)erx sin sx−2rserx cos sx erx √ = sin sx 2 2 2 2 (r −s +q) +(2rs) (r 2 −s2 +q)2 +(2rs)2



− tan−1

2rs r 2 −s2 +q

3r 2 −q 4r 3 −4qr   2r P  (x) + (r2+q) (x) + · · · 2 P (x) − (r 2 +q)3 P r 2 +q  k k−1 k−3 k k−5 2 −(k )r q+( )r q −··· k−1 (1 )r (k−1) 3 5 P (x) + · · · + · · · + (−1) 2 +q)k−1 (r  3s2 +q 5s4 +10s2 q+q 2 iv  sin sx P (x) + · · · P (x) − (−s 2 +q)2 P (x) + (−s2 +q) (−s2 +q)4 erx r 2 +q

P (x) −

(2k+1 )s2k +(2k+1 )s2k−2 q+(2k+1 )s2k−4 q2 +··· (2k) 3 5 +(−1)k 1 P (x) + · · · (−s2 +q)2k   2 2P (x) 4s +4q  s cos sx − (−s (x) + · · · − (−s 2 +q) 2 +q)3 P (−s2 +q) 2k 2k−2 2k 2k−4 s + s q+··· ( ) ( ) k+1 1 (2k−1) 3 P (x) + · · · +(−1) (−s2 +q)2k−1

Table IV: (D 2 +b2 )y = R 29. sin bx∗ 30. P (x) sin bx∗

bx − x cos 2b sin bx (2b)2

P (x) −

P  (x) (2b)2

+

P iv (x) (2b)4

 − ··· −

cos bx 2b



P (x) −

P  (x) (2b)2

 + · · · dx

* For cos sx in R replace “sin” by “cos” and “cos” by “− sin” in yp.

A-61



DIFFERENTIAL EQUATIONS (Continued) Table V: (D 2 + pD+q)y = R R 31. erx

yp

erx r 2 +pr+q (q−s2 ) sin sx−ps cos sx (q−s2 )2 +(ps)2

32. sin sx∗

1 q

33. P (x) 34. erx sin sx∗ 35. P (x)erx



= √

1 (q−s2 )2 +(ps)2

 sin sx − tan−1

ps q−s2



3 p2 −q  P (x) − p −2pq P  (x) + · · · q2 q3  n−1 n−2 n−2 n−4 2 n p −(1 )p q+(2 )p q −··· (n) P (x) +(−1)n qn 2

P (x) − pq P  (x) +

Replace p by p + 2r, q by q+pr+r in formula 32 and multiply by erx . Replace p by p + 2r, q by q+pr+r2 in formula 33 and multiply by erx .

Table VI: (D − b)(D − a)y = R



   a a2 − s2 b b 2 − s2  P (x) P (x) + − − a2 + s2 b 2 + s2 (a2 + s2 )2 (b2 + s2 )2    b3 − 3bs2 a3 − 3as2  − (x) + · · · P + (a2 + s2 )3 (b2 + s2 )3     cos sx s 2as s 2bs  + P (x) + P (x) − 2 − 2 2 2 2 2 2 2 2 2 b−a a +s b +s (a + s ) (b + s )  †  3a2 s − s2 3b2 s − s3  + − (x) + · · · P (a2 + s2 )3 (b2 + s2 )3

sin sx b−a

36. P (x) sin sx∗

37. P (x)erx sin sx∗ Replace and multiply by erx .   a by a–r, b by b − r in formula 36  ax P (x) P (x) P (x) P (n) (x) e + (b−a) 38. P (x)eax P (x) dx + (b−a) 2 + (b−a)3 + · · · + (b−a)n+1 a−b * For cos sx in R replace “sin” by “cos” and “cos” by “-sin” in yp . † For additional terms, com e with formula 6.

Table VII: (D 2 − 2aD + a2 + b2 )y = R R

yp ∗









P  (x)  3a (s−b)−(s−b)  + 3a[a(s+b)−(s+b) P − (x) + · · · 3 3 2 +(s+b)2 ]   [a22+(s−b)22]  a −(s+b) a2 −(s−b)2 cos sx a a − 2b P (x) + P  (x) − − 2 2 2 2 2 2 2 2 2 2 a +(s+b) a +(s−b) [a +(s+b) ] [a +(s−b) ]  2   † 2 3 2 + a[a2−3a(s+b) P  (x) + · · · − a[a2−3a(s−b) +(s+b)2 ]3 +(s−b)2 ]3 rx 40. P (x)erx sin sx∗  Replace a by a − r in formula  39 and multiply by e . 39. P (x) sin sx

sin sx 2b

s+b

a2 +(s+b)2

−

s−b



a2 +(s−b)2

P (x) +

2

3

2a(s+b)

[a2 +(s+b)2 ]2 2

−

2a(s−b)



[a2 +(s−b)2 ]2

3

iv P  (x) eax + P b4(x) − · · · b2 P (x) − b2 eax sin sx eax sin sx∗ −s2ax +b2 eax sin bx∗ − xe 2bcos bx   P  (x) P iv (x) eax sin bx P (x) − P (x)eax sin bx∗ + − · · · (2b)2 (2b)2 (2b)4   iv P  (x) eax cos bx − 2b P (x) − (2b)2 + P(2b)(x) 4 − · · · dx

41. P (x)eax

42. 43. 44.

* For cos sx in R replace “sin’ by “cos’ and “cos” by “-sin” in yp . † For additional terms, com e with formula 6.

A-62

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DIFFERENTIAL EQUATIONS (Continued) Table VIII: f (D)y = [D n + an−1 D n−1 + · · · + a1 D + a0 ]y = R R 45. erx 46. sin sx∗

yp

erx f (r) [a0 −a2 s2 +a4 s4 −··· ] sin sx−[a1 s−a3 s3 +a5 s5 +··· ] cos sx [a0 −a2 s2 +a4 s4 −··· ]2 +[a1 s−a3 s3 +a5 s5 −··· ]2

Table IX: f (D 2 )y = R 47. sin sx∗

sin sx f (−s2 )

=

sin sx a0 −a2 s2 +···±s2n

Table X: (D − a)n y = R 48. erx 49. sin sx∗

erx (r−a)n n n−2 2  4  n−4 4 (−1)n n s + n a s − · · · ] sin sx (a2 +s2 )2 {[a − 2 a n n−3 3 n n−1 +[ s − s + · · · ] cos sx} a a 1  n P  (x) 3 n+1 P  (x) n+2 P  (x) (−1)n P (x) + 1 a + 2 an a2 + a2 3 rx

 50. P (x) + ··· 51. erx sin sx∗ Replace a by a − r in formula 49 and multiply by e . Replace a by a − r in formula 50 and multiply by erx . 52. erx P (x) n  n + 1   53. P (x) sin sx∗ (−1)n sin sx[An P (x) + An+1 P (x) + An+2 P (x) + n + 2

1

2



(x) + · · · ] 3  n n + 1  n   Bn+1 P (x) + Bn+2 P (x) + + (−1) cos sx[Bn P (x) + 1 2 n + 2  Bn+3 P (x) + · · · ] 3   k−2 2 k k−4 4 s + 4 a s − ··· ak − k a a2 − s2 2 a A1 = 2 , A2 = , . . . , Ak = 2 2 2 2 a +s (a + s ) (a2 + s2 )k   k−3 3 k k−1 s− k s + ··· a 2as 1 a 3 a B1 = 2 , B = , . . . , B = 2 k a + s2 (a2 + s2 )2 (a2 + s2 )k rx An+3 P

54. erx sin sx∗

Replace a by a − r in formula 53 and multiply by e .

55. eax P (x)

eax

56. P (x)eax sin sx∗

···

(−1)

n−1 2

P (x) dxn  ax

e sn

sin sx

 P  (x) n s n−1

−

n+2 P  (x) n−1

s3

+

n+4 P v (x) n−1

s5

− ···



 n+1 P  (x) n+3 P iv (x) P (x) − n−1 + − · · · (n odd) 2 4 n−1 s s   n n+1 P  (x) n+3 P iv (x) (−1) 2 eax sin sx n−1 P (x) − n−1 s2 + n−1 s4 − · · · n−1 sn   n n+2 P  (x) n+4 P v (x) (−1) 2 eax cos sx  n  P  (x) + − n−1 + n−1 s5 − · · · (n even) sn s n−1 s3 n+1

+

(−1) 2

eax cos sx sn



n−1 n−1



* For cos sx in Rreplace “sin” by “cos” and “cos” by “− sin” in yp.

Table XI: (D − a)n f (D)y = R 57. eax

xn n!

ax

· fe(a) *For cos sxin Rreplace “sin” by “cos” and “cos” by “− sin” in yp.

A-63

DIFFERENTIAL EQUATIONS (Continued) Table XII: (D 2 + q)n y = R R 58. erx 59. sin sx∗ 60. P (x)

yp erx /(r2 + q)n sin sx/(q − s2 )n  n P  (x) n+1 P iv (x) n+2 P vi (x) 1 + − + · · · P (x) − n 2 2 3 q 1 2 3 q q q erx (A2 +B 2 )n

61. erx sin sx∗

" n   n n n−2 2 n−4 4 A − A B + A B − · · · sin sx 2 4   #  n n n−1 n−3 3 − A B− A B + · · · cos sx 1 3

2

2

A = r − s + q,

B = 2rs

Table XIII: (D 2 + b2 )n y = R 62. sin bx∗

n

cos bx (−1)n+1/2 xn!(2b) n

n

sin bx (n odd), (−1)n/2 xn!(2b) n

(n even)

Table XIV: (D n − q)y = R n erx /(r  − q)

63. erx

(n)

64. P (x)

− 1q P (x) P

65. sin sx∗

− q sin sx+(−1) q 2 +s2n

rx

66. e

sin sx∗

(x)

q n−1 2

Ae

rx

+

P (2n) (x) q2

sn cos sx

+ ···



(n odd),

rx



rx sin sr−Be cos sx = √ e2 2 sin A2 +B 2 A  n n n−2 2 n +B r − 2 r  s + 4 rn−4 s4 −  n n−1 s − n3 rn−3 s3 + · · · r 1

sx

sin sx (−s2 )n/2 −q − tan−1 B A

(n even)

 · · · − q, A= B= *For cos sx in R replace “sin” by “cos” and cos by “− sin” in yp.

Table XV: (Dx + mDy )z = R R zp eax+by 67. eax+by a+mb f (u)du 68. f (ax + by) ∫ a+mb , u = ax + by 69. f (y − mx) xf (y − mx) 70. φ(x, y)f (y − mx) f (y − mx) ∫ φ(x, a + mx)dx (a = y − mx after integration)

Table XVI: (Dx + mDy − k)z = R 71. eax+by 72. sin(ax + by)∗ 73. eαx+βy sin(ax + by)∗ 74. exk f (ax + by) 75. f (y − mx) 76. p(x)f (y − mx) 77. ekx f (y − mx)

eax+by a+mb−k sin(ax+by) − (a+bm) cos(ax+by)+k (a+bm)2 +k2

Replace k in 72 by k − α − mβand multiply by eαx+βy 

ekx f (u)du , u = ax + by a+mb − f (y−mx) k   − k1 f (y − mx) p(x) + p k(x) kx

+

p (x) k2

+ ··· +

p(n) (x) kn



xe f (y − mx) *For cos(ax + by) replace “sin” by “cos” and “cos” by “-sin” in zp . k+r ∂ ∂ ; Dy = ∂y ; Dxk Dyr = ∂∂k ∂ r Dx = ∂x x

A-64

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y

DIFFERENTIAL EQUATIONS (Continued) Table XVII: (Dz + mDy )n z = R R 78. eax+by

zp eax+by n (a+mb)   ··· f (u)dun , (a+mb)n xn f (y − mx) n! 

79. f (ax + by) 80. f (y − mx) 81. φ(x, y)f (y + mx)

f (y − mx)

u = ax + by ···



φ(x, a + mx)dxn (a = y − mx after integration)

Table XVIII:(Dx + mDy − k)n z = R eax+by (a+mb−k)n (−1)n f (y−mx) kn     (−1)n f (y − mx) p(x) + n1 p k(x) kn ekx ∫ ∫ ··· ∫ f (u)dun , u = ax + by (a+mb)n xn kx e f (y − mx) n!

82. eax+by 83. f (y − mx) 84. P (x)f (y − mx) 85. ekz f (ax + by) 86. ekx f (y − mx)

+

n+1 p (x) 2

k3

+

n+2 p (x) 3

k3

+ ···

Table XIX: [Dxn + a1 Dxn−1 Dy + a2 Dxn−2 Dy2 + · · · + an Dyn ]z = R 87. eax+by

eax+by a + a1 an−1 b + a2 an−2 b2 + · · · + an bn 

88. f (ax + by)

 · · · f (u)dun , (u = ax + by) an + a1 an−1 b + a2 an−2 b2 + · · · + an bn Table XX: F (Dx , Dy )z = R

89. eax+by

eax+by F (a,b)

Table XXI:F (Dx2 , Dx Dy , Dy2 )z = R 90. sin(ax + by)∗

sin(ax+by) F (−a2 ,−ab,−b2 )

*For cos(ax + by)replace “sin ” by “cos”, and “cos” by “-sin” in zp .

A-65



DIFFERENTIAL EQUATIONS (Continued) DIFFERENTIAL EQUATIONS G(v) dv yF (xy) dx + x G(xy) dy = 0 ln x = +c v{G(v) − F (v)} where v = xy. If G(v) = F (v), the solution is xy = c. Linear, homogeneous, second order equation d2 y dy + cy = 0 +b dx2 dx b, c are real constants

Let m1 , m2 be the roots of m2 + bm + c = 0. Then there are 3 cases: Case 1.

m1 , m2 real and distinct:

Case 2.

y = c1 em1 x + c2 em2 x m1 , m2 real and equal:

Case 3.

y = c1 em1 x + c2 xem1 x m1 = p + qi, m2 = p − qi :

y = epx (c1 cos qx + c2 sin qx) √ where p = −b/2, q = 4c − b2 /2 Linear, nonhomogeneous, second order equation d2 y dy + cy = R(x) +b 2 dx dx b, c are real constants

There are 3 cases corresponding to those immediately above: Case 1.

Case 2.

Case 3.

y = c1 em1 x + c2 em2 x em1 x + e−m1 x R(x) dx m1 − m 2 em2 x e−m2 x R(x) dx + m2 − m 1 y = c1 em1 x + c2 xem1 x + xem1 x e−m1 x R(x) dx − em1 x xe−m1 x R(x) dx y = epx (c1 cos qx + c2 sin qx) epx sin qx + e−px R(x) cos qx dx q epx cos qx − e−px R(x) sin qx dx q

A-66

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DIFFERENTIAL EQUATIONS (Continued) Putting x = et , the equation becomes

Euler or Cauchy equation x2

dy d2 y + bx + cy = S(x) dx dx

dy d2 y + (b − 1) + cy = S(et ) dt2 dt and can then be solved as a linear second order equation. y = c1 Jn (λx) + c2 Yn (λx)

Bessel’s equation x2

dy d2 y +x + (λ2 x2 − n2 )y = 0 dx2 dx y = x−p c1 Jq/r

Transformed Bessel’s equation x2

dy d2 y + (2p + 1)x + (α2 x2r + β 2 )y = 0 dx2 dx

where q =

  α ! xr + c2 Yq/r xr r r


p2 − β 2 .

y = c1 Pn (x) + c2 Qn (x)

Legendre’s equation (1 − x2 )

α

d2 y dy − 2x + n(n + 1)y = 0 dx2 dx

Differential equation

Method of solution f1 (x) g2 (y) dx + dy = c f2 (x) g1 (y)



Separation of variables f1 (x)g1 (y) dx + f2 (x)g2 (y) dy = 0 Exact equation M (x, y) dx + N (x, y) dy = 0 where ∂M/∂y = ∂N/∂x

Linear first order equation dy + P (x)y = Q(x) dx Bernoulli’s equation dy + P (x)y = Q(x)y n dx Homogeneous equation y dy =F dx x Reducible to homogeneous (a1 x + b1 y + c1 )dx +(a2 x + b2 y + c2 ) dy = 0 b1 a1 = a2 b2 Reducible to separable (a1 x + b1 y + c1 ) dx +(a2 x + b2 y + c2 ) dy = 0 b1 a1 = a2 b2



M ∂x+



n−

∂ ∂y



 M ∂x dy = c

where ∂x indicates that the integration is to be performed with respect to x keeping y constant.   ye P dx = Qe P dx dx + c ve(1−n)



P dx =

where v = y

1−n





Qe(1−n)



P dx

dx + c

if n = 1, the solution is

ln y = (Q  −dvP ) dx + c ln x = F (v)−v +c where v = y/x.If F (v) = v, the solution is y = cx Set u = a1 x + b1 y + c1 v = a 2 x + b 2 y + c2 Eliminate x and y and the equation becomes homogenous

Set u = a1 x + b1 y Eliminate x or y and equation becomes separable

A-67

FOURIER SERIES 1. If f (x) is a bounded periodic function of period 2L (i.e. f x + 2L) = f (x), and satisfies the Dirichlet conditions: (a) In any period f (x) is continuous, except possibly for a finite number of jump discontinuities. (b) In any period f (x) has only a finite number of maxima and minima. Then f (x) may be represented by the Fourier series a0   nπx nπx  an cos + + bn sin 2 L L n=1 ∞

where an and bn are as determined below. This series will converge to f (x) at every point where f (x) is continuous, and to f (x+ ) + f (x− ) 2 (i.e., the average of the left-hand and right-hand limits) at every point where f (x) has a jump discontinuity.

an

=

bn

=

1 L 1 L



L

f (x) cos

nπx dx, L

n = 0, 1, 2, 3, . . . ,

f (x) sin

nπx dx, L

n = 1, 2, 3, . . .

−L L −L

we may also write 1 an = L



α+2L

α

nπx 1 f (x) cos dx and bn = L L



α+2L

f (x) sin α

nπx dx L

where α is any real number. Thus if α = 0,

an bn

= =

1 L 1 L



2L

f (x) cos

nπx dx, L

n = 0, 1, 2, 3, . . . ,

f (x) sin

nπx dx, L

n = 1, 2, 3, . . .

0



0

2L

2. If in addition to the restrictions in (1), f (x) is an even function (i.e., f (−x) = f (x)), then the Fourier series reduces to ∞ nπx a0  + an cos 2 L n=1 That is, bn = 0. In this case, a simpler formula for an is 2 L nπx an = f (x) cos dx, n = 0, 1, 2, 3, . . . L 0 L

A-68

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3. If in addition to the restrictions in (1), f (x) is an odd function (i.e., f (−x) = −f (x)), then the Fourier series reduces to ∞  nπx bn sin L n=1 That is, an = 0. In this case, a simpler formula for the bn is bn =

2 L



L

f (x) sin 0

nπx dx, L

n = 1, 2, 3, . . .

4. If in addition to the restrictions in (2) above, f (x) = −f (L − x), then an will be 0 for all even values of n, including n = 0. Thus in this case, the expansion reduces to ∞ 

a2m−1 cos

m=1

(2m − 1)πx L

5. If in addition to the restrictions in (3) above, f (x) = f (L − x), then bn will be 0 for all even values of n. Thus in this case, the expansion reduces to ∞ 

b2m−1 sin

m=1

(2m − 1)πx L

(The series in (4) and (5) are known as odd-harmonic series, since only the odd harmonics appear. Similar rules may be stated for even-harmonic series, but when a series appears in the even-harmonic form, it means that 2L has not been taken as the smallest period of f (x). Since any integral multiple of a period is also a period, series obtained in this way will also work, but in general computation is simplified if 2L is taken to be the smallest period.) 6. If we write the Euler definitions for cos θ and sin θ, we obtain the complex form of the Fourier Series known either as the “Complex Fourier Series” or the “Exponential Fourier Series” of f (x). It is represented as n=+∞ 1  f (x) = cn eiωn x 2 n=−∞ where 1 cn = L



L

f (x) e−iωn x dx,

−L

n = 0, ±1, ±2, ±3, . . .

with ωn = nπ L for n = 0, ±1, ±2, . . . The set of coefficients cn is often referred to as the Fourier spectrum. 7. If both sine and cosine terms are present and if f (x) is of period 2L and expandable by a Fourier series, it can be represented as   nπx a0  cn sin + + φn , 2 L n=1 ∞

f (x) =

an = cn sin φn ,

bn = cn cos φn ,

where cn =

A-69




 a2n

+

b2n ,

φn = arctan

an bn



It can also be represented as   nπx a0  f (x) = + + φn , cn cos 2 L n=1 ∞

an = cn cos φn ,

where

bn = −cn sin φn ,

cn =




a2n + b2n ,

  bn φn = arctan − an

where φn is chosen so as to make an , bn , and cn hold. 8. The following table of trigonometric identities should be helpful for developing Fourier series. n 0 (−1)n

sin nπ cos nπ ∗ sin nπ 2 ∗ cos nπ 2 sin nπ 4

neven 0 +1 0 (−1)n/2

*A useful formula for sin sin

nπ 2

nodd 0 −1 (−1)(n−1)/2 0 √ (n2 +4n+11)/8 2 (−1) 2

and cos

nπ 2

nπ (i)n+1 = [(−1)n − 1] 2 2

n/2 odd 0 +1 0 −1 (−1)(n−2)/4

n/2 even 0 +1 0 +1 0

is given by

and

cos

nπ (i)n = [(−1)n + 1], 2 2

where i2 = −1.

AUXILIARY FORMULAS FOR FOURIER SERIES  4 1 3πx 1 5πx πx + sin + sin + ··· [0 < x < k] sin π k 3 k 5 k  2k 1 2πx 1 3πx πx x= − sin + sin − ··· [−k < x < k] sin π k 2 k 3 k  k 3πx 5πx πx 4k 1 1 x = − 2 cos + 2 cos + 2 cos + ··· [0 < x < k] 2 π k 3 k 5 k     2k2 3πx π2 πx 4 π2 2πx 4 π2 − sin − sin + − 3 sin x2 = 3 π 1 1 k 2 k 3 3 k   2 2 π 5πx 4πx 4 π − sin + − 3 sin + ··· [0 < x < k] 4 k 5 5 k  k2 2πx 3πx 4πx πx 4k2 1 1 1 − 2 cos − 2 cos + 2 cos − 2 cos + ··· x2 = 3 π k 2 k 3 k 4 k 1=

[−k < x < k]

1− 1 22 1 1− 2 2 1 1+ 2 3 1 1 + 2 22 4 1−

1 1 1 + − 3 5 7 1 1 + 2 + 2 3 4 1 1 + 2 − 2 3 4 1 1 + 2 − 2 5 7 1 1 + 2 + 2 6 8

+ ··· = + ··· = + ··· = + ··· = + ··· =

A-70

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π 4 π2 6 π2 12 π2 8 π2 24

FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS f (x) =

f (x) =

4 π

2 π

 n=1,3,5...

∞  n=1

∞ 

c 2 f (x)= L +π

f (x) =

f (x) =

f (x) =

f (x) =

f (x) =

A-71

n=1

∞ 

2 L

2 π

1 2

n=1

∞  n=1

−

8 π2

1 2

1 n

(−1)n n

(−1)n n

sin nπ 2

 nπx cos nπc L − 1 sin L

nπc L

n=1,3,5,...

∞  n=1

1 n

nπx L

sin nπx L

sin nπx L



n=1,3,5,...

cos

sin( 12 nπc/L) 1 2 nπc/L

(−1)n+1 n

4 π2

1 π



sin



−

sin nπx L

1 n2

cos nπx L

(−1)(n−1)/2 n2

sin nπx L

sin nπx L

FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS (Continued) f (x) = 12 (1 + a) +

f (x) =

f (x) =

f (x) =

f (x) π4

2 π

1 2

2 π

∞  n=1

−

n=1

∞  n=1

f (x) =

f (x) =

f (x) =

9 π2

1 n

1 π

(−1)n n

sin

∞  n=1

32 3π 2

n=1

1 2

nπ 4

1 n2

∞ 

+



(−1)n−1 n

4 π 2 (1−2a)

∞ 

1+

n=1

n=1,3,5,...

sin nπa sin

nπ 3

sin

sin ωt −

sin

nπ 4

2 π

1 n2

sin

nπx ; L



nπx ; L



n=2,4,6,...

a=

a=







c 2L

c 2L

a=

nπx ; L

 sin nπa sin

nπx ; L

nπx ; L

sin



cos nπa cos

1+(−1)n nπ(1−2a)

1+

cos nπa − 1] cos nπx ; L   c a = 2L

1 [(−1)n n2

sin nπa nπ(1−a)





sin

1 n2

∞ 

2 π 2 (1−a)

c 2L



nπx ; L



a=



c 2L

a=







a=

c 2L

1 n2 −1

cos nωt

Extracted from graphs and formulas, pages 372, 373, Differential Equations in Engineering Problems, Salvadori and Schwarz, published by Prentice-Hall, Inc.,1954.

A-72

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c 2L



THE FOURIER TRANSFORMS For a piecewise continuous function F (x) over a finite interval 0 ≤ x ≤ π; the finite Fourier cosine transform of F (x) is π fc (n) = F (x) cos nx dx (n = 0, 1, 2, . . . ) 0

If x ranges over the interval 0 ≤ x ≤ L, the substitution x = πx/L allows the use of this definition, also. The inverse transform is written. x 1 2  F (x) = fc (0) − fc (n) cos nx (0 < x < π) π π n=1 where F (x) = formula

F (x+')+F (x−') . 2

We observe that F (x+) = F (x−) = F (x) at points of continuity. The

π

fc(2) (n) =

F  (x) cos nx dx (1)

0

= −n2 fc (n) − F  (0) + (−1)n F  (π) makes the finite Fourier cosine transform useful in certain boundary value problems. Analogously, the finite Fourier sine transform of F (x) is π fs (n) = F (x) sin nx dx (n = 1, 2, 3, . . . ) 0

and F (x) = Corresponding to (1) we have fs(2) (n)

∞ 2  fs (n) sin nx (0 < x < π) π n=1

=

π

F  (x) sin nx dx

(2)

0

=

−n2 fs (n) − n F (0) − n(−1)n F (π)

Fourier Transforms If F (x) is defined for x ≤ 0 and is piecewise continuous over any finite interval, and if absolutely convergent, then

x 2 fc (α) = F (x) cos(αx) dx π 0

x 0

F (x) dx is

is the Fourier cosine transform of F (x). Furthermore,

x 2 F (x) = fc (α) cos(αx) dα. π 0 If limx→∞ dn F/dxn = 0, then an important property of the Fourier cosine transform is

x  2r  d F 2 (2r) fc (α) = cos(αx) dx π 0 dx2r

r−1 2  (−1)n a2r−2n−1 α2n + (−1)r α2r fc (α) =− π n=0 where limx→∞ dr F/dxr = ar, makes it useful in the solution of many problems. Under the same conditions.

x 2 fs (α) = F (x) sin(αx) dx π 0

A-73

(3)

defines the Fourier sine transform of F (x), and

F (x) = Corresponding to (3) we have

fs(2r) (α)

2 π

= =

−



2 π



x

fs (α) sin(αx) dα 0

∞

d2r F sin(αx) dx dx2r 0 r 2  (−1)n α2n−1 a2r−2n + (−1)r−1 α2r fs (α) π n=1

∞ F (x)dx is absolutely convergent, then Similarly, if F (x) is defined for −∞ < x < ∞, and if ∫−∞ ∞ 1 F (x)eiax dx f (α) = √ 2π −∞

is the Fourier transform of F (x), and 1 F (x) = √ 2π Also, if



∞

f (α)e−iax dα

−∞

 n  d F  lim  n  = 0 (n = 1, 2, . . . , r − 1) |x|→∞ dx

then 1 f (r) (α) = √ 2π



∞

F (r) (x)eiαx dx = (−iα)r f (α)

−∞

Finite Sine Transforms π

1 fs (n) =

0

n+1

2 (−1)

fs (n) F (x) sin nx dx (n = 1, 2, . . . )

F (x) F (x) F (π − x)

fs (n)

3

1 n

π−x π

4

(−1)n+1 n

x π

5

1−(−1)n n

6

2 n2

7

(−1)n+1 n3

x(π 2 −x2 ) 6π

8

1−(−1)n n3

x(π−x) 2

9

π 2 (−1)n−1 n

sin

1    x

nπ 2

10 π(−1)n 11

n [1 n2 +c2

12

n n2 +c2



when 0 < x < π/2

  π − x when π/2 < x < π

−

6 n3

2[1−(−1)n ] n3

−

π2 n

x2



x3

− (−1)n ecπ ]

ecx sinh c(π−x) sinh cπ

A-74

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(4)

fs (n) (k = 0, 1, 2, . . . )

n

13

n2 −k2

14

  π2

when n = m

 0

when n = m

F (x) sin k(π−x) sin kπ

(m = 1, 2, . . . )

n [1 − (−1)n cos kπ] n2 − k 2 (k = 1, 2, . . . )  n  [1 − (−1)n+m ]  n2 −m2 when n = m = 1, 2, . . .   0 when n = m n (k =  0, 1, 2, . . . ) 2 2 2 (n −k ) bn (|b| ≤ 1) n 1−(−1)n n b (|b| ≤ 1) n

15

16 17 18 19

sin mx cos kx

cos mx k(π−x) π sin kx − x cos 2k sin kπ 2k sin2 kπ 2 b sin x arctan 1−b cos x π 2 sin x arctan 2b1−b 2 π

Finite Cosine Transforms 1 2 3

fc (n) π fc (n) = 0 F (x) cos nx dx (n = 0, 1, 2, . . . ) (−1)n fc (n) 0 when n = 1, 2, · · · ;fc (0) = π

4

2 n

sin

nπ ; fc (0) 2

=0

n

2

; fc (0) = π2 5 − 1−(−1) n2 2 (−1)n 6 ; fc (0) = π6 n2 7 n12 ; fc (0) = 0 n n 4 8 3π 2 (−1) − 6 1−(−1) ; fc (0) = π4 n2 n4 (−1)n ec π−1 9 n2 +c2 1 10 n2 +c 2 k 11 [( − 1)n cos πk − 1] n2 − k 2 (k = 0, 1, 2, · · · ) (−1)n+m −1 12 ; fc (m) = 0 (m = 1, 2, · · · ) n2 −m2 1 (k = 0, 1, 2, . . . ) 13 n2 −k 2 14 0 when n = 1, 2, · · · ; π fc (m) = (m = 1, 2, · · · ) 2

F (x) F (x) F (π − x) 1 $ 1 when 0 < x < π/2 −1 when π/2 < x < π x x2 2π (π−x)2 2π 3

−

π 6

x

1 cx e c coshc(π−x) csinhcπ

sin kx

1 m

sin mx k(π−x) − cosk sin kπ cos mx

Fourier Sine Transforms

1 2 3

F (x) $ 1 (0 < x < a) 0 (x > a)

e−x

5

xe−x

7



xp−1 (0 < p < 1) $ sin x (0 < x < a) 0 (x > a)

4 6



2

cos sin

fs (α) 2 π

 1−cos α  α

2 Γ(p) π αp

√1 2π



 

sin

sin[a(1−α)] 1−α

2 α π 1+α2 −α2 /2

/2

pπ 2



−

sin[a(1+α)] 1+α



αe   2  2 ∗ √ 2 2 2 sin α2 c α2 − cos α2 S α2  2  2 ∗ √  2 2 2 cos α2 C α2 + sin α2 S α2

2

x 2

x2 2

A-75

Here C(y) and S(y) are the Fresnel integrals: y 1 1 √ cos t dt, C(y) = √ t 2π 0

1 S(y) = √ 2π



y

0

1 √ sin t dt t

*More extensive tables of the Fourier sine and cosine transforms can be found in Fritz Oberhettinger, Tabellen zur-Fourier Transformation, Springer, 1957.

Fourier Cosine Transforms F (x) $ 1 (0 < x < a) 0 (x > a)

1

 

xp−1 (0 < p < 1) $ cos x (0 < x < a) 0 (x > a)

2 3 4

e−x

5

e−x

2

6

cos

7

sin

x2 2

2 sin aα π α 2 Γ(p) π αp

√1 2π



 

cos

e



cos cos



pπ 2

sin[a(1−α)] 1−α

2 1 π 1+α2 1 −α /2

/2

x2 2

fc (α)

+



α2 2

−

π 4

α2 2

+

π 4

sin[a(1+α)] 1+α



 

Fourier Transforms F (x) 1

2

0

"

4

eiwx (p < x < q) 0 (x " −cx+iwx < p, x > q) (x > 0) e (c > 0) 0 (x < 0) 2 e−px R(p) > 0

5

cos px2

6

sin px2

7

|x|−p

8

−a|x| e√ |x|

9 10

cosh ax cosh πx sinh ax sinh πx $

2 3

12 13 14

15

ip(w+α) −eiq(w+α) √i e (w+α) 2π

i √ 2π(w+α+ic) 2 √1 e−α /4p 2p  2 √1 cos α 4p 2p

(0 < p < 1)

√ 2 2 √ a +x ]

 

cos

2 cos 2 cosh 2 π cosh α+cos a 1 sin a √ 2π cosh α+cos a


π

(|x| > a)

"

sin[b

"

π 4

√ 2 2 (a +α )+a √ 2 2  a +α a α

a −x

0



−

α2 + π4 4p  pπ 2 Γ(1−p) sin 2 (1−p) π |α|  √1 2p

(−π < a < π) (−π < a < π) √ 1 (|x| < a) 2 2

11

f (α) |α| < a |α| > a

"
π

sin ax x

a2 +x2

pn (x) (|x| < 1)  0 √ (|x| > 1) 2 2  cos[b √ a −x ] (|x| < a) a2 −x2  0 (|x| > a)  √ a2 −x2 ]  cosh[b √ (|x| < a) a2 −x2  0 (|x| > a)

2

J0 (aα) 0 √ J (a b2 − α2 ) 0 2


π

in √ J 1 (α) α n+ 2


π 2


π 2

A-76

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√ J0 (a a2 + b2 ) √ J0 (a α2 − b2 )

(|α| > b) (|α| < b)

*More extensive tables of Fourier transforms can be found in W . Magnus and F . Oberhettinger, Formulas and Theorems of the Special Functions of Mathematical Physics. Chelsea, 1949, 116–120. The following functions appear among the entries of the tables on transforms. Function Ei(x) Si(x) Ci(x) erf(x) erfc(x) Ln (x)

Definition  x ev dv; or sometimes defined as −x v  x −v = x e v dv −Ei(−x) x sin v dv v 0 x

cos v v

dv; or sometimes defined as negative of this integral  x −v2 √2 e dv π 0 ∞ 2 1− erf(x) = √2π x e−v dv x n e d (xn e−x ), n = 0, 1, . . . n! dxn x

Name Sine, Cosine, and Exponential Integral tables pages 548–556 Sine, Cosine, and Exponential Integral tables pages 548–556 Sine, Cosine, and Exponential Integral tables pages 548–556 Error function Complementary function to error function

Laguerre polynomial of degree n

n

SERIES EXPANSION The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated it is to be understood that the series converges for all finite values of x.

(x + y)n

=

(1 ± x)n

=

(1 ± x)−n

=

(1 ± x)−1

=

BINOMIAL SERIES n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3 xn + nxn−1 y + y + y + · · · (y 2 < x2 ) x x 2! 3! n(n − 1)(n − 2)x3 n(n − 1)x2 ± + · · · (x2 < 1) 1 ± nx + 2! 3! n(n + 1)x2 n(n + 1)(n + 2)x3 1 ∓ nx + ∓ + · · · (x2 < 1) 2! 3! 1 ∓ x + x2 ∓ x3 + x4 ∓ x5 + · · · (x2 < 1)

−2

=

1 ∓ 2x + 3x2 ∓ 4x3 + 5x4 ∓ 6x5 + · · ·

(1 ± x)

(x2 < 1)

REVERSION OF SERIES Let a series be represented by y = a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + · · · with a1
= 0. The coefficients of the series x = A 1 y + A 2 y 2 + A 3 y 3 + A4 y 4 + · · · are A1 =

1 a1

A2 = − A4 =

A5

=

A6

=

A7

=

a2 a31

A3 =

1 (2a22 − a1 a3 ) a51

1 (5a1 a2 a3 − a21 a4 − 5a32 ) a71

1 (6a21 a2 a4 + 3a21 a23 + 14a42 − a31 a5 − 21a1 a22 a3 ) a91 1 (7a31 a2 a5 + 7a31 a3 a4 + 84a1 a32 a3 − a41 a6 − 28a21 a22 a4 − 28a21 a2 a33 − 42a52 ) a11 1 1 (8a41 a2 a6 + 8a41 a3 a5 + 4a41 a24 + 120a21 a32 a4 + 180a21 a22 a23 + 132a62 − a51 a7 a13 1 −36a31 a22 a5 − 72a31 a2 a3 a4 − 12a31 a33 − 330a1 a42 a3 )

A-77

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TAYLOR SERIES 1. f (x) =f (a) + (x − a)f  (a) +

(x − a)2  (x − a)3  f (a) + f (a) 2! 3!

(x − a)n (n) f (a) + · · · n! (Increment form) + ··· +

(Taylor’s Series)

h2  f (x) + 2! x2 = f (h) + xf  (h) + f  (h) + 2!

2. f (x + h) = f (x) + hf  (x) +

h3  f (x) + · · · 3! x3  f (h) + · · · 3!

3. If f (x) is a function possessing derivatives of all orders throughout the interval a ≤ x ≤ b, then there is a value X, with a < X < b, such that f (b) = f (a) + (b − a)f  (a) +

(b − a)2  (b − a)n−1 (n−1) (b − a)n (n) f (a) + · · · + f f (X) (a) + 2! (n − 1)! n!

f (a + h) = f (a) + hf  (a) +

h2  hn−1 (n−1) hn (n) (a) + f (a) + · · · + f f (a + θh) 2! (n − 1)! n!

where b = a + h and 0 < θ < 1. Or f (x) = f (a) + (x − a)f  (a) +

(x − a)2  f (n−1) (a) f (a) + · · · + (x − a)n−1 + Rn , 2! (n − 1)!

where

f (n) [a + θ · (x − a)] (x − a)n , 0 < θ < 1. n! The above forms are known as Taylor’s series with the remainder term. Rn =

4. Taylor’s series for a function of two variables   ∂ ∂ ∂f (x, y) ∂f (x, y) If h +k f (x, y) = h +k ; ∂x ∂y ∂x ∂y  2 2 ∂ 2 f (x, y) ∂ ∂ 2 f (x, y) ∂ 2 ∂ f (x, y) h f (x, y) = h2 + 2hk +k + k ∂x ∂y ∂x2 ∂x∂y ∂y 2 y=b  n  ∂ ∂ etc., and if h ∂x + k ∂y f (x, y) where the bar and subscripts means that after differx=a entiation we are to replace x by a and y by b, then y=b y=b    n   ∂ 1 ∂ ∂ ∂ f (a+h, b+k) = f (a, b)+ h +· · ·+ f (x, y) +· · · +k f (x, y) h +k ∂x ∂y n! ∂x ∂y x=a x=a MACLAURIN SERIES f (n−1) (0) x2  x3 f (x) = f (0) + xf (0) + f (0) + f  (0) + · · · + xn−1 + Rn , 2! 3! (n − 1)! 

where Rn =

xn f (n) (θx) , n!

A-78

0 < θ < 1.

EXPONENTIAL SERIES 1 1 1 1 + + + + ··· 1! 2! 3! 4! x3 x4 x2 + + + ··· ex = 1 + x + 2! 3! 4! (x loge a)2 (x loge a)3 ax = 1 + x loge a + + + ··· 2! 3!  (x − a)2 (x − a)3 x a e = e 1 + (x − a) + + + ··· 2! 3! e=1+

LOGARITHMIC   SERIES  3 x−1 1 x−1 2 + + 13 x−1 + ··· x 2 x x (x− 1) − 12 (x − 1)2 + 13 (x − 1)3 − · · ·  3  5 1 x−1 1 x−1 loge x = 2 x−1 + + + ··· x+1 3 x+1 5 x+1

(x > 12 ) (2 ≥ x > 0)

loge x = loge x =

(x > 0)

loge (1 + x) = x − 12 x2 + 13 x3 − 14 x4 + (−1 < x ≤ 1)  ··· loge (n + 1) − loge (n − 1) = 2 n1 + 3n1 3+ 5n1 5 + · · ·  3 x x loge (a + x) = loge a + 2 2a+x + 13 2a+x  5 x + 15 2a+x + ··· (a > 0, −a < x < +∞)   3 5 2n−1 1+x = 2 x + x3 + x5 + · · · + x2n−1 + · · · −1 < x < 1 loge 1−x loge x =

loge a +

(x−a) a

−

(x−a)2 2a2

+

(x−a)3 3a3

− +···

0 < x ≤ 2a

TRIGONOMETRIC SERIES 3 5 7 sin x = x − x3! + x5! − x7! + · · · (all real values of x) 2 4 6 cos x = 1 − x2! + x4! − x6! + · · · (all real values of x) n−1 2n 3 5 7 9 (22n −1)B2n 2n−1 tan x = x + x3 + 2x + 17x + 62x + · · · + (−1) 2(2n)! x + ··· , 15 315 2835   2 x2 < π4 and Bn represents the nth Bernoulli number x3 45 

5

n+1 2n

7

(−1) 2 x − 2x B2n x2n−1 − · · · , 945 − 4725 − · · · − (2n)!  2 2 th x < π and Bn represents the n Bernoulli number n 2 5 4 61 6 277 8 2n x + 720 x + 8064 x + · · · + (−1) + ··· , sec x = 1 + x2 + 24 (2n)! E2n x   2 π 2 th x < 4 and En represents the n Euler number 7 31 127 csc x = x1 + x6 + 360 x3 + 15,120 x5 + 604,800 x7 + · · ·

cot x =

1 x

−

x 3

+ (−1)

−

n+1

2(22n−1 −1)

 (2n)! x2 < π 2

B2n x2n−1 + · · · ,  and Bn represents the nth Bernoulli number

    2 2 2 sin x = x 1 − πx2 1 − 22xπ2 1 − 32xπ2 · · ·     2 4x2 4x2 cos x = 1 − 4x 1 − 1 − ··· 2 2 2 2 2 π 3 π 5 π x 1·3 5 1·3·5 7 sin−1 x = x + 2·3 x + 2·4·6·7 x + ···  + 2·4·5  3 7 π x 1·3 5 −1 + · · · cos x = 2 − x + 2·3 + 2·4·5 x + 1·3·5x 2·4·6·7 3

tan−1 x tan−1 x tan−1 x cot−1 x

3

5

7

= x − x3 + x5 − x7 + · · · = π2 − x1 + 3x1 3 − 5x1 5 + 7x1 7 − · · · = − π2 − x1 + 3x1 3 − 5x1 5 + 7x1 7 − · · · 3 5 7 = π2 − x + x3 − x5 + x7 − · · · A-79

TeamLRN

(x2 < ∞) 

x2 < 1, − π2

(x2 < ∞)  < sin−1 x < π2

(x2 < 1, 0 < cos−1 x < π) (x2 < 1) (x > 1) (x < −1) (x2 < 1)

loge sin x =

loge x − 2

loge cos x = − x2 − loge tan x = esin x

=

cos x

e

=

etan x

=

sin x =

x2 6

4

x 12

−

−

x4 180

6

x 45

−

x6 2835 − · · · 17x8 2520 − · · ·

−

4 62x6 loge x + x3 + 7x 90 + 2835 + · · · 2 4 8x5 3x6 56x7 1 + x + x2! − 3x 4! − 5! − 6! + 7! 2 4 6 31x e 1 − x2! + 4x 4! − 6! + · · ·

2 2 (x < π2 ) x2 < π4   2 x2 < π4

2

+

3x3 3!



9x4 37x5 4! + 5! + · · · 2 sin a + (x − a) cos a − (x−a) sin a 2! (x−a)3 (x−a)4 − 3! cos a + 4! sin a + · · ·

1+x+

x2 2!

+ ···

+

x2 <

π2 4



1 2

v, |V −1 /V V ˆ V0 v 1 V2 V 21 V r rV=Vr r r r r s1

|=v . /v v


= 0) V1 -V2 − V2 . 1

1 +V2 .

,

23

V1 + V2 = V2 + V1 (r + s)V1 = rV1 + sV1 ; r(V1 + V2 ) = rV1 + rV2 V1 + (V2 + V3 ) = (V1 + V2 ) + V3 = V1 + V2 + V3 . V = rV1 + sV2

12

12

ABOP V = rV1 + (1 − r)V2

A-80

4!

sin a + · · ·

VECTOR ANALYSIS Definitions Any quantity which is completely determined by its magnitude is called a scalar. Examples of such are mass, density, temperature, etc. Any quantity which is completely determined by its magnitude and direction is called a vector. Examples of such are velocity, acceleration, force, etc. A vector quantity is represented by a directed line segment, the length of which represents the magnitude of the vector. A vector quantity is usually represented by a boldfaced letter such as V. Two vectors V1 and V2 are equal to one another if they have equal magnitudes and are acting in the same directions. A negative vector, written as -V, is one which acts in the opposite direction to V, but is of equal magnitude to it. If we represent the magnitude of V by v, we write |V| = v. A vector parallel to V, but equal to the reciprocal of its magnitude is written as V−1 or as 1/V. The unit vector V/v (when v
= 0) is that vector which has the same direction as V, but has a ˆ ). magnitude of unity (sometimes represented as V0 or v Vector Algebra The vector sum of V1 and V2 is represented by V1 +V2 . The vector sum of V1 and -V2 , or the difference of the vector V2 from V1 is represented by V1 − V2 . If r is a scalar, then rV=Vr, and represents a vector r times the magnitude of V, in the same direction as V if r is positive, and in the opposite direction if r is negative. If r and s are scalars, V1 , V2 , V3 , vectors, then the following rules of scalars and vectors hold: V1 + V2 = V2 + V1 (r + s)V1 = rV1 + sV1 ; r(V1 + V2 ) = rV1 + rV2 V1 + (V2 + V3 ) = (V1 + V2 ) + V3 = V1 + V2 + V3 Vectors in Space A plane is described by two distinct vectors V1 and V2 . Should these vectors not intersect each other, then one is displaced parallel to itself until they do (Fig. 1). Any other vector V lying in this plane is given by V = rV1 + sV2 A position vector specifies the position in space of a point relative to a fixed origin. If therefore V1 and V2 are the position vectors of the points A and B, relative to the origin O, then any point P on the line AB has a position vector V given by V = rV1 + (1 − r)V2

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The scalar “r” can be taken as the metric representation of P since r = 0 implies P = B and r = 1 implies P = A (Fig. 2). If P divides the line AB in the ratio r:s then     r s V= V1 + V2 r+s r+s

Figure 1. Figure 2. The vectors V1 , V2 , V3 , . . . ,Vn are said to be linearly dependent if there exist scalars r1 , r2 , r3 , . . . ,rn , not all zero, such that r1 V 1 + r2 V 2 + · · · + rn V n = 0 A vector V is linearly dependent upon the set of vectors V1 , V2 , V3 , . . . ,Vn if V = r 1 V 1 + r 2 V 2 + r3 V 3 + · · · + r n V n Three vectors are linearly dependent if and only if they are co-planar. All points in space can be uniquely determined by linear dependence upon three base vectors i.e., three vectors any one of which is linearly independent of the other two. The simplest set of base vectors are the unit vectors along the coordinate Ox, Oy and Oz axes. These are usually designated by i, j and k respectively. If V is a vector in space, and a, b and c are the respective magnitudes of the projections of the vector along the axes then V = ai + bj + ck and v=




a2 + b2 + c2

and the direction cosines of V are cos α = a/v,

cos β = b/v,

cos γ = c/v.

The law of addition yields V1 + V2 = (a1 + a2 )i + (b1 + b2 )j + (c1 + c2 )k The Scalar, Dot, or Inner Product of Two Vectors This product is represented as V1 · V2 and is defined to be equal to v1 v2 cos θ, where θ is the angle from V1 to V2 , i.e., V1 · V2 = v1 v2 cos θ The following rules apply for this product: V1 · V2 = a1 a2 + b1 b2 + c1 c2 = V2 · V1 It should be noted that this verifies that scalar multiplication is commutative. (V1 + V2 ) · V3 = V1 · V3 + V2 · V3 V1 · (V2 + V3 ) = V1 · V2 + V1 · V3 A-81

If V1 is perpendicular to V2 then V1 · V2 = 0, and if V1 is parallel to V2 then V1 · V2 = v1 v2 = rw12 In particular i · i = j · j = k · k = 1, and i·j=j·k=k·i=0 The Vector or Cross Product of Two Vectors This product is represented as V1 × V2 and is defined to be equal to v1 v2 (sin θ)1, where θ is the angle from V1 to V2 and 1 is a unit vector perpendicular to the plane of V1 and V2 and so directed that a right-handed screw driven in the direction of 1 would carry V1 into V2 , i.e., V1 × V2 = v1 v2 (sin θ)1 |V1 × V2 | V1 · V2 The following rules apply for vector products:

and tan θ =

V1 × V2

= −V2 × V1

V1 × (V2 + V3 )

= V1 × V2 + V1 × V3

(V1 + V2 ) × V3

= V1 × V3 + V2 × V3

V1 × (V2 × V3 ) = V2 (V3 · V1 ) − V3 (V1 · V2 ) i × i = j × j = k × k = 0 (the zero vector) i × j = k,

j × k = i,

k×i

= j

If V1 = a1 i + b1 j + c1 k, V2 = a2 i + b2 j + c2 k, and V3 = a3 i + b3 j + c3 k, then    i j k   V1 × V2 =  a1 b1 c1  = (b1 c2 − b2 c1 )i + (c1 a2 − c2 a1 )j + (a1 b2 − a2 b1 )k  a2 b2 c2  It should be noted that, since V1 × V2 = −V2 × V1 , the vector product is not commutative. Scalar Triple Product There is only one possible interpretation of the expression V1 · V2 × V3 and that is V1 · (V2 × V3 ) which is obviously a scalar. Further V1 · (V2 × V3 ) = (V1 × V2 ) · V3 = V2 · (V3 × V1 )   a1 b1 c1    = a2 b2 c2  a3 b3 c3  = r1 r2 r3 cos φ sin θ, Where θ is the angle between V2 and V3 and φ is the angle between V1 and the normal to the plane of V2 and V3 . This product is called the scalar triple product and is written as [V1 V2 V3 ]. The determinant indicates that it can be considered as the volume of the parallelepiped whose three determining edges are V1 , V2 and V3 . It also follows that cyclic permutation of the subscripts does not change the value of the scalar triple product so that [V1 V2 V3 ] = [V2 V3 V1 ] = [V3 V1 V2 ] but

[V1 V2 V3 ] = −[V2 V1 V3 ]

etc.

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and

[V1 V1 V2 ] ≡ 0

etc.

Given three non-coplanar reference vectors V1 , V2 and V3 , the reciprocal system is given by V1∗ , V2∗ and V3∗ , where 1 = v1 v1∗ = v2 v2∗ = v3 v3∗ 0 = v1 v2∗ = v1 v3∗ = v2 v1∗ etc. V2 × V3 V3 × V1 V1∗ = , V2∗ = , [V1 V2 V3 ] [V1 V2 V3 ]

V3∗ =

V1 × V2 [V1 V2 V3 ]

The system i, j, k is its own reciprocal. Vector Triple Product The product V1 × (V2 × V3 ) defines the vector triple product. Obviously, in this case, the brackets are vital to the definition. V1 × (V2 × V3 ) = (V1 · V3 )V2 − (V1 · V2 )V3   i j k   b c a 1 1 1     =     c2 a2   a2 b2 c b 2 2         b3 c3   c3 a3   a3 b3

   

       

i.e. it is a vector, perpendicular to V1 , lying in the plane of V2 , V3 . Similarly     i j k           b1 c1   c1 a1   a1 b1         (V1 × V2 ) × V3 =     c2 a2   a2 b2     b2 c2   b3 c3 a3 V1 × (V2 × V3 ) + V2 × (V3 × V1 ) + V3 × (V1 × V2 ) ≡ 0 If V1 × (V2 × V3 ) = (V1 × V2 ) × V3 then V1 , V2 , V3 form an orthogonal set. Thus i, j, k form an orthogonal set. Geometry of the Plane, Straight Line and Sphere The position vectors of the fixed points A, B, C, D relative to O are V1 , V2 , V3 , V4 and the position vector of the variable point P is V. The vector form of the equation of the straight line through A parallel to V2 is V = V1 + rV2 or

(V − V1 ) = rV2

or

(V − V1 ) × V2 = 0

while that of the plane through A perpendicular to V2 is (V − V1 ) · V2 = 0 The equation of the line AB is V = rV1 + (1 − r)V2 and those of the bisectors of the angles between V1 and V2 are   V1 V2 or V=r ± v1 v2 V = r(ˆ v1 ± v ˆ2 ) The perpendicular from C to the line through A parallel to V2 has as its equation ˆ2 · (V1 − V3 )ˆ v2 . V = V1 − V 3 − v A-83

The condition for the intersection of the two lines, V = V1 + rV3 and V = V2 + sV4 is [(V1 − V2 )V3 V4 ] = 0. The common perpendicular to the above two lines is the line of intersection of the two planes [(V − V1 )V3 (V3 × V4 )] = 0

and

[(V − V2 )V4 (V3 × V4 )] = 0

and the length of this perpendicular is [(V1 − V2 )V3 V4 ] . |V3 × V4 | The equation of the line perpendicular to the plane ABC is V = V1 × V2 + V2 × V3 + V3 × V1 and the distance of the plane from the origin is [V1 V2 V3 ] . |(V2 − V1 ) × (V3 − V1 )| In general the vector equation V · V2 = r defines the plane which is perpendicular to V2 , and the perpendicular distance from A to this plane is r − V1 · V2 v2 The distance from A, measured along a line parallel to V3 , is r − V1 · V2 V2 · v ˆ3

or

r − V1 · V2 v2 cos θ

where θ is the angle between V2 and V3 . (If this plane contains the point C then r = V3 · V2 and if it passes through the origin then r = 0.) Given two planes V · V1 = r V · V2 = s then any plane through the line of intersection of these two planes is given by V · (V1 + λV2 ) = r + λs where λ is a scalar parameter. In particular λ = ±v1 /v2 yields the equation of the two planes bisecting the angle between the given planes. The plane through A parallel to the plane of V2 , V3 is V = V1 + rV2 + sV3 or

(V − V1 ) · V2 × V3 = 0

or

[VV2 V3 ] − [V1 V2 V3 ] = 0

so that the expansion in rectangular Cartesian coordinates yields (where V ≡ xi + yj + zk):   (x − a1 ) (y − b1 ) (z − c1 )    a2 b2 c2  = 0   a3 b3 c3 

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which is obviously the usual linear equation in x, y, and z. The plane through AB parallel to V3 is given by [(V − V1 )(V1 − V2 )V3 ] = 0 or

[VV2 V3 ] − [VV1 V3 ] − [V1 V2 V3 ] = 0

The plane through the three points A, B and C is V = V1 + s(V2 − V1 ) + t(V3 − V1 ) or V = rV1 + sV2 + tV3 (r + s + t ≡ 1) or

[(V − V1 )(V1 − V2 )(V2 − V3 )] = 0

or

[VV1 V2 ] + [VV2 V3 ] + [VV3 V1 ] − [V1 V2 V3 ] = 0

For four points A, B, C, D to be coplanar, then rV1 + sV2 + tV3 + uV4 ≡ 0 ≡ r + s + t + u The following formulae relate to a sphere when the vectors are taken to lie in three dimensional space and to a circle when the space is two dimensional. For a circle in three dimensions take the intersection of the sphere with a plane. The equation of a sphere with center O and radius OA is V · V = v12 or

(notV = V1 )

(V − V1 ) · (V + V1 ) = 0

while that of a sphere with center B radius v1 is (V − V2 ) · (V − V2 ) = v12 or V · (V − 2V2 ) = v12 − v22 If the above sphere passes through the origin then V · (V − 2V2 ) = 0 (note that in two dimensional polar coordinates this is simply) r = 2a · cos θ while in three dimensional Cartesian coordinates it is x2 + y 2 + z 2 − 2 (a2 x + b2 y + c2 x) = 0. The equation of a sphere having the points A and B as the extremities of a diameter is (V − V1 ) · (V − V2 ) = 0. The square of the length of the tangent from C to the sphere with center B and radius v1 is given by (V3 − V2 ) · (V3 − V2 ) = v12 The condition that the plane V · V3 = s is tangential to the sphere (V − V2 ) · (V − V2 ) = v12 is (s − V3 · V2 ) · (s − V3 · V2 ) = v12 v32 . A-85

The equation of the tangent plane at D, on the surface of sphere (V − V2 ) · (V − V2 ) = v12 , is (V − V4 ) · (V4 − V2 ) = 0 or V · V4 − V2 · (V + V4 ) = v12 − v22 The condition that the two circles (V − V2 ) · (V − V2 ) = v12 and (V − V4 ) · (V − V4 ) = v32 intersect orthogonally is clearly (V2 − V4 ) · (V2 − V4 ) = v12 + v32 The polar plane of D with respect to the circle (V − V2 ) · (V − V2 ) = v12 is V · V4 − V2 · (V + V4 ) = v12 − v22 Any sphere through the intersection of the two spheres (V − V2 ) · (V − V2 ) = v12 and (V − V4 ) · (V − V4 ) = v32 is given by (V − V2 ) · (V − V2 ) + λ(V − V4 ) · (V − V4 ) = v12 + λv32 while the radical plane of two such spheres is 1 V · (V2 − V4 ) = − (v12 − v22 − v32 + v42 ) 2 Differentiation of Vectors If V1 = a1 i + b1 j + c1 k, and V2 = a2 i + b2 j + c2 k, and if V1 and V2 are functions of the scalar t, then d dV1 dV2 (V1 + V2 + · · · ) = + + ··· dt dt dt dV1 da1 db1 dc1 = i+ j+ k, etc dt dt dt dt d dV1 dV2 (V1 · V2 ) = · V2 + V1 · dt dt dt dV1 dV2 d (V1 × V2 ) = × V2 + V1 × dt dt dt dV dv V· =v· dt dt In particular, if V is a vector of constant length then the right hand side of the last equation is identically zero showing that V is perpendicular to its derivative. The derivatives of the triple products are 

      dV2 dV3 V2 V3 + V1 V3 + V1 V2 and dt dt         dV1 dV2 dV3 d {V1 × (V2 × V3 )} = × (V2 × V3 ) + V1 × × V3 + V1 × V2 × dt dt dt dt d [V1 V2 V3 ] = dt

dV1 dt



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Geometry of Curves in Space s = the length of arc, measured from some fixed point on the curve (Fig. 3). V1 = the position vector of the point A on the curve V1 + δV1 = the position vector of the point P in the neighborhood of A ˆ t =the unit tangent to the curve at the point A, measured in the direction of s increasing. The normal plane is that plane which is perpendicular to the unit tangent. The principal normal is defined as the intersection of the normal plane with the plane defined by V1 and V1 +δV1 in the limit as δV1 − 0. n ˆ = the unit normal (principal) at the point A. The plane defined by ˆ t and n ˆ is called the osculating plane (alternatively plane of curvature or local plane). ρ = the radius of curvature at A. δθ=the angle subtended at the origin by δV1 . κ=

dθ 1 = ds ρ

ˆ =the unit binormal i.e. the unit vector which is parallel to ˆ b t×n ˆ at the point A: λ = the torsion of the curve at A

Figure 3. Frenet’s Formulae: dˆ t = κˆ n ds dˆ n ˆ = −κˆ t + λb ds ˆ db = −λˆ n ds The following formulae are also applicable: 1 ˆ Unit tangent t = dV ds Equation of the tangent (V − V1 ) × ˆ t=0 1d2 V1 Unit normal n ˆ = κds2 Equation of the normal plane (V − V1 ) · ˆ t=0 Equation of the normal (V − V1 ) × n ˆ=0 ˆ =ˆ Unit binormal b t×n ˆ ˆ=0 Equation of the binormal (V − V1 ) × b ˆ or V = V1 + ub

or

V = V1 + qˆ t

or

V = V1 + rˆ n

2

Equation of the osculating plane:

d V1 1 or V = V1 + w dV ds × ds2 ˆ [(V − V1 )tn ˆ] = 0  1   d2 V1  or (V − V1 ) dV =0 ds ds2

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Differential Operators—Rectangular Coordinates dS =

∂S ∂S ∂S · dx + · dy + · dz ∂x ∂y ∂z

By definition ∂ ∂ ∂ ∇ ≡ del ≡ i ∂x + j ∂y + k ∂z 2 2 ∂ ∂ ∇2 ≡ Laplacian ≡ ∂x 2 + ∂y 2 +

∂2 ∂z 2

∂S ∂S If S is a scalar function, then ∇S ≡ grad S ≡ ∂S dx i + dy j + dz k Grad S defines both the direction and magnitude of the maximum rate of increase of S at any point. Hence the name gradient and also its vectorial nature. ∇S is independent of the choice of rectangular coordinates.

Figure 4.

∇S =

∂S n ˆ ∂n

(5)

where n ˆ is the unit normal to the surface S =constant, in the direction of S increasing. The total derivative of S at a point having the position vector V is given by (Fig. 4) ∂S n ˆ · dV ∂n = dV · ∇S

dS =

and the directional derivative of Sin the direction of U is U · ∇S = U · (∇S) = (U · ∇)S Similarly the directional derivative of the vector V in the direction of U is (U · ∇)V The distributive law holds for finding a gradient. Thus if S and T are scalar functions ∇(S + T ) = ∇S + ∇T The associative law becomes the rule for differentiating a product: ∇(ST ) = S∇T + T ∇S If V is a vector function with the magnitudes of the components parallel to the three coordinate axes Vx , Vy , Vz , then ∂Vx ∂Vy ∂Vz ∇ · V ≡ div V ≡ + + ∂x ∂y ∂z

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The divergence obeys the distributive law. Thus, if V and U are vector functions, then ∇ · (V + U) = ∇ · V + ∇ · U ∇ · (SV) = (∇S) · V + S(∇ · V) ∇ · (U × V) = V · (∇ × U) − U · (∇ × V) As with the gradient of a scalar, the divergence of a vector is invariant under a transformation from one set of rectangular coordinates to another. ∇ × V ≡ curl V ( sometimes ∇ΛV or rot V)       ∂Vx ∂Vy ∂Vz ∂Vx ∂Vx ∂Vy ≡ − i+ − j+ − k ∂y ∂z ∂z ∂x ∂x ∂y    i j k   ∂ ∂ ∂  =  ∂x ∂y ∂z   Vx Vy Vz  The curl (or rotation) of a vector is a vector which is invariant under a transformation from one set of rectangular coordinates to another. ∇ × (U + V) = ∇ × U + ∇ × V ∇ × (SV) = (∇S) × V + S(∇ × V) ∇ × (U × V) = (V · ∇)U − (U · ∇)V + U(∇ · V) − V(∇ · U) If V = Vx i + Vy j + Vz k then ∇ · V = ∇Vx · i + ∇Vy · j + ∇Vz · k and ∇ × V = ∇Vx × i + ∇Vy × j + ∇Vz × k The operator ∇ can be used more than once. The possibilities where ∇ is used twice are: ∇ · (∇θ) ≡ div grad θ ∇ × (∇θ) ≡ curl grad θ ∇(∇ · V) ≡ grad div V ∇ · (∇ × V) ≡ div curl V ∇ × (∇ × V) ≡ curl curl V Thus, if S is a scalar and V is a vector: div grad S ≡ ∇ · (∇S) ≡ Laplacian S ≡ ∇2 S ≡

∂2S ∂2S ∂2S + + ∂x2 ∂y 2 ∂z 2

curl grad S ≡ 0 curl curl V ≡ grad div V − ∇2 V; div curl V ≡ 0 Taylor’s expansion in three dimensions can be written f (V + ε) = eε·∇ f (V)

where V = xi + yj + zk and ε = hi + lj + mk

A-89

Orthogonal Curvilinear Coordinates If at a point P there exist three uniform point functions u, vand w so that the surfaces u =const., v =const., and w =const., intersect in three distinct curves through P then the surfaces are called the coordinate surfaces through P . The three lines of intersection are referred to as the coordinate lines and their tangents a, b, and c as the coordinate axes. When the coordinate axes form an orthogonal set the system is said to define orthogonal curvilinear coordinates at P . Consider an infinitesimal volume enclosed by the surfaces u, v, w, u + du, v + dv, and w + dw (Fig. 5).

Figure 5. The surface P RS ≡ u = constant, and the face of the curvilinear figure immediately opposite this is u + du =constant, etc. In terms of these surface constants P = P (u, v, w) Q = Q(u + du, v, w)

and P Q = h1 du

R = R(u, v + dv, w)

and P R = h2 dv

S = S(u, v, w + dw)

and P S = h3 dw

where h1 , h2 , and h3 are functions of u, v, and w. • In rectangular Cartesians i, j, k h1 = 1, h2 = 1, h3 = 1. ˆ ∂ ˆ ∂ ∂ b Φ ˆ c ∂ ˆ a ∂ ˆ∂ . =i , = , =k h1 ∂u ∂x h2 ∂v r ∂φ h3 ∂w ∂z ˆΦ ˆ • In cylindrical Cartesians ˆ r, θ, h1 = 1, h2 = 1, h3 = 1. ˆ ∂ ˆ ∂ ∂ b Φ ˆ c ∂ ˆ a ∂ ˆ∂ . = rˆ , = , =k h1 ∂u ∂r h2 ∂v r ∂φ h3 ∂w ∂z ˆΦ ˆ • In spherical coordinates ˆ r, θ, h1 = 1, ˆ a ∂ ∂ =ˆ r , h1 ∂u ∂r

h2 = r, ˆ ∂ b ∂ Φ = , h2 ∂v r ∂θ

h3 = r sin θ ˆ ˆ c ∂ Φ ∂ = h3 ∂w r sin θ ∂φ

The general expressions for grad, div and curl together with those for ∇2 and the directional derivative are, in orthogonal curvilinear coordinates, given by:

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∇S = (V · ∇)S = ∇·V = ∇×V =

∇2 S =

ˆ ∂S b ˆ c ∂S ˆ a ∂S + + h1 ∂u h2 ∂v h3 ∂w V1 ∂S V2 ∂S V3 ∂S + + h1 ∂u h ∂v h3 ∂w " 2 # 1 ∂ ∂ ∂ (h2 h3 V1 ) + (h3 h1 V2 ) + (h1 h2 V3 ) . h1 h2 h3 ∂u ∂v ∂w " # # ˆ " ∂ ˆ a ∂ b ∂ ∂ (h3 V3 ) − (h2 V2 ) + (h1 V1 ) − (h3 V3 ) h2 h3 ∂v ∂w h3 h1 ∂w ∂u " # ˆ c ∂ ∂ + (h2 V2 ) − (h1 V1 ) h1 h2 ∂u ∂v "      # 1 ∂ h3 h1 ∂S ∂ ∂ h2 h3 ∂S h1 h2 ∂S + + h1 h2 h3 ∂u h1 ∂u ∂v h2 ∂v ∂w h3 ∂w FORMULAS OF VECTOR ANALYSIS

Rectangular coordinates

Cylindrical coordinates

Conversion to rectangular coordinates

x = r cos ϕ

Gradient . . .

∇φ =

Divergence ...

∇·A =

Curl . . .

Laplacian ...

∂φ ∂x i

+

∂Ax ∂x

∂φ ∂y j

+

  i  ∂ ∇ × A =  ∂x  Ax

∇2 φ =

∂2 φ ∂x2

+

∂Ay ∂y

+

k

∂ ∂y

∂ ∂z

Ay

Az

+

∂φ ∂r r

∇·A= z + ∂A ∂z

∂Az ∂z

j

∂2 φ ∂y 2

∇φ =

∂φ ∂z k

+

     

y = r sin ϕ z = z

1 ∂φ r ∂ϕ Φ

+

1 ∂(rAr ) r ∂r

 1   r∂ r ∇ × A =  ∂r  Ar

∇2 φ =

∂2 φ ∂z 2

1 ∂ r ∂r



+

1 rk ∂ ∂z

rAϕ

Az

r ∂φ ∂r

 +

2

(c)

F (c)



(∇φ) · ˆ t ds =

(c)

dφ (c)

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z = r cos θ

∂φ ∂r r

+

1 ∂φ r ∂θ θ

2 1 ∂(r Ar ) ∂r r2 ∂A ϕ 1 + r sin θ ∂ϕ

     

+

∂φ 1 r sin θ ∂ϕ Φ

+

∂(Aθ sin θ) 1 r sin θ ∂θ

 r Φ θ   r2 sin θ r sin θ r ∂ ∂ ∂ ∇ × A =  ∂r ∂θ ∂ϕ  A rAθ rAϕ sin θ r

1 ∂2 φ r 2 ∂ϕ2

∇2 φ =



1 ∂ r 2 ∂φ ∂r r 2 ∂r ∂2 φ 1 + r2 sin2 θ ∂ϕ2

ds ) dS= n ˆ



y = r sin ϕ sin θ

∇·A=

1 ∂Aϕ r ∂ϕ

∂ ∂ϕ

x = r cos ϕ sin θ

∇φ =

∂φ ∂z k

Φ

F ·ˆ t ds = = ∇φ

+

+ ∂∂zφ 2

ˆ ˆ F ds= ˆ t . sC. S. V t C P n

F

Spherical coordinates

 +

1 ∂ r 2 sin θ ∂θ



     

sin θ ∂φ ∂θ



dS )

∂ϕ2

Transformation of Integrals If 1. 2. 3. 4. 5. 6. 7. 8. then

s is the distance along a curve “C” in space and is measured from some fixed point. S is a surface area V is a volume contained by a specified surface ˆ t =the unit tangent to C at the point Pn ˆ =the unit outward pointing normal F is some vector function ds is the vector element of curve (= ˆ t ds ) dS is the vector element of surface (= n ˆ dS ) ˆ F · t ds = F (c)

and when F = ∇φ

(c)



(∇φ) · ˆ t ds =

(c)

dφ (c)

Gauss’ Theorem (Green’s Theorem) When S defines a closed region having a volume V : (∇ · F) dV = (F · n ˆ ) dS = F · dS (v) (s)



(∇φ) dV = (∇ × F) dV =

also

φn ˆ dS

(v)

and

(s)



(v)

(s)

(ˆ n × F) dS

(s)

Stokes’ Theorem When C is closed and bounds the open surface S: n ˆ · (∇ × F) dS = F · ds (s) (c)





(ˆ n × ∇φ) dS =

also

φ ds

(s) (c)

Green’s Theorem



(∇φ · ∇θ) dS =

φn ˆ · (∇θ) dS =

φ(∇2 θ) dV (v)

(s)

(s)



θ·n ˆ (∇φ)dS =

=

φ(∇2 θ) dV (v)

m

(s)

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5

MOMENT OF INERTIA FOR VARIOUS BODIES OF MASS The mass of the body is indicated by m Body

Axis

Moment of inertia

Uniform thin rod

m l3

The rectangular sheet, sides a and b

Normal to the length, at one end Normal to the length, at the center Through the center parallel to b

Thin rectangular sheet, sides a and b Thin circular sheet of radius r

Through the center perpendicular to the sheet Normal to the plate through the center

ma

Thin circular sheet of radius r

Along any diameter

m r4

Thin circular ring. Radii r1 and r2

Through center normal to plane of ring

m

2 +r 2 r1 2 2

Thin circular ring. Radii r1 and r2

Any diameter

m

2 +r 2 r1 2 4

Rectangular parallelepiped, edges a, b, and c Sphere, radius r

Through center perpendicular to face ab, (parallel to edge c) Any diameter

ma

Spherical shell, external radius, r1 , internal radius r2

Any diameter

Uniform thin rod

2

2

l m 12 2

m a12 2 +b2 12

2

m r2

2

2 +b2 12

m 25 r 2 m 25

5 −r 5 ) (r1 2 3 −r 3 ) (r1 2

Body

Axis

Moment of inertia

Spherical shell, very thin, mean radius, r Right circular cylinder of radius r, length l Right circular cylinder of radius r, length l

Any diameter

m 23 r 2

The longitudinal axis of the solid Transverse diameter

m r2

Hollow circular cylinder, length l, radii r1 and r2 Thin cylindrical shell, length l, mean radius, r

The longitudinal axis of the figure The longitudinal axis of the figure

m

Hollow circular cylinder, length l, radii r1 and r2

Transverse diameter

m

Hollow circular cylinder, length l, very thin, mean radius

Transverse diameter

m

Elliptic cylinder, length l, transverse semiaxes a and b

Longitudinal axis

m

Right cone, altitude h, radius of base r

Axis of the figure

3 2 m 10 r

Spheroid of revolution, equatorial radius r

Polar axis

m 2r5

Ellipsoid, axes 2a, 2b, 2c

Axis 2a

m (b

A-92

2

 m

r2 4

+

l2 12



2 +r 2 ) (r1 2 2

mr2  



2 +r 2 r1 2 4 r2 2

+

a2 +b2 4

2

2 +c2 ) 5

+ l2 12



l2 12





SPECIAL FUNCTIONS Bessel Functions 1. Bessel’s differential equation for a real variable x is x2

d2 y dy +x + (x2 − n2 )y = 0 2 dx dx

2. When n is not an integer, two independent solutions of the equation are Jn (x), J−n (x), where Jn (x) =

∞  k=0

x (−1)k k!Γ(n + k + 1) 2

n+2k

3. If n is an integer Jn (x) = (−1)n Jn (x), where # " xn x4 x6 x2 Jn (x) = n + 4 + 6 + ... 1− 2 2 n! 2 · 1!(n + 1) 2 · 2!(n + 1) (n + 2) 2 · 3!(n + 1) (n + 2) (n + 3) 4. For n = 0 and n = 1, this formula becomes x2 22 (1!)2 x x3 2 − 23 ·1!2!

J0 (x) = 1 −

+

J1 (x) =

+

x4 24 (2!)2 x5 25 ·2!3!

− −

x6 x8 26 (3!)2 + 28 (4!)2 − · · · x7 x9 27 ·3!4! + 29 ·4!5! − . . .

5. When x is large and positive, the following asymptotic series may be used 1   π π ! 2 2 − Q0 (x) sin x − P0 (x) cos x − πx 4 4  &  2  12 %   3π 3π J1 (x) = πx P1 (x) cos x − 4 − Q1 (x) sin x − 4 

J0 (x) =

where 1 2 · 32 12 · 32 · 52 · 72 12 · 32 · 52 · 72 · 92 · 112 + − + ··· 2!(8x)2 4!(8x)4 6!(8x)6 12 12 ·32 ·52 12 ·32 ·52 ·72 ·92 Q0 (x) ∼ − 1!8x + 3!(8x)3 − + −··· 5!(8x)5

P0 (x) ∼ 1 −

2 2 2 2 2 12 ·3·5 12 ·32 ·52 ·7·9 ·7 ·9 ·11·13 + 1 ·3 ·56!(8x) 6 2!(8x)2 − 4!(8x)4 2 2 2 2 2 2 1·3 1 ·3 ·5·7 1 ·3 ·5 ·7 ·9·11 − + − · · · 3 5 1!8x 3!(8x) 5!(8x)

P1 (x) ∼ 1 + Q1 (x) ∼

− +···

[In P1 (x) the signs alternate from+to-after the first term] 6. The zeros of J0 (x) and J1 (x). If j0s and j1s are the sth zeros of J0 (x) and J1 (x) respectively, and if a = 4s − 1, b = 4s + 1 # " 1 62 15, 116 12, 554, 474 8, 368, 654, 292 2 j0,s ∼ πa 1 + 2 2 − 4 4 + − + − +··· 10 a10 4 % π a 3π a 15π 6 a6 105π 8 a8 315π & 3,902,418 895,167,324 4716 j1,s ∼ 14 πb 1 − π26b2 + π46b4 − 5π 6 b6 + 35π 8 b8 − 35π 10 b10 + · · · & % s+1 3 2 (−1) 2 56 9664 J1 (j0,s ) ∼ 1 − 3π4 a4 + 5π6 a6 − 7,381,280 1 21π 8 a8 + · · · πa 23 & s 2 % 19,584 2,466,720 J0 (j1,s ) ∼ (−1) 12 1 + π24 4 b4 − 10π 6 b6 + 7π 8 b8 − · · · πb 2

A-93

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SPECIAL FUNCTIONS 7. Table of zeros for J0 (x) and J1 (x) Define {αn , βn } by J1 (αn ) = 0 and J0 (βn ) = 0. Roots αn 2.4048 5.5201 8.6537 11.7915 14.9309 18.0711 21.2116

J1 (αn ) 0.5191 -0.3403 0.2715 -0.2325 0.2065 -0.1877 0.1733

Roots β n 0.0000 3.8317 7.0156 10.1735 13.3237 16.4706 19.6159

J0 (βn ) 1.0000 -0.4028 0.3001 -0.2497 0.2184 -0.1965 0.1801

8. Recurrence formulas 2n Jn (x) nJn (x) + xJn (x) = xJn−1 (x) x  Jn−1 (x) − Jn+1 (x) = 2Jn (x) nJn (x) − xJn (x) = xJn+1 (x)

Jn−1 (x) + Jn+1 (x) =

(k)

9. If Jn is written for Jn (x) and Jn relationships are important (r)

J0 (2) J0 (3) J0 (4) J0

is written for

(r−1)



J 12 (x) =

2

πx 

J− 12 (x) = Jn+ 32 (x) = Jn− 12 (x) =

sin x

2 πx cos x 1 d 1 −xn+ 2 dx {x−(n+ 2 ) Jn+ 12 (x)} 1 1 d x−(n+ 2 ) dx {xn+ 2 Jn+ 12 (x)}

 πx  12

n 0

2

x3

2

− cos x

 − 1 sin x − x3 cos x    − x6 sin x − x152 − 1 cos x etc.

  15

 πx  12

Jn+ 12 (x) sin x

sin x x

1 2

then the following derivative

= −J1 = −J0 + x1 J1 = 12 (J2 − J0 )   = x1 J0 + 1 − x22 J1 = 14 (−J3 + 3J1 )     = 1 − x32 J0 − x2 − x63 J1 = 18 (J4 − 4J2 + 3J0 ), etc.

10. Half order Bessel functions

3

dk {Jn (x)}, dxk



3 x2

−

 15 x3

J−(n+ 12 ) (x) cos x

3 x2

− cosx x − sin x  − 1 cos x + x3 sin x

−

6 x



cos x −

 15 x2

 − 1 sin x

11. Additional solutions to Bessel’s equation are Yn (x) (also called Weber’s function, and sometimes denoted by Nn (x)) (1) (2) Hn (x) and Hn (x) (also called Hankel functions) These solutions are defined as follows    Jn (x) cos (nπ) − J−n (x) sin(nπ) Yn (x) =  −v (x)  lim Jv (x) cos(vπ)−J sin(vπ) v→n

n

not an integer n

A-94

an integer

(1)

Hn (x) = Jn (x) + iYn (x) (2) Hn (x) = Jn (x) − iYn (x)

SPECIAL FUNCTIONS The additional properties of these functions may all be derived from the above relations and the known properties of Jn (x). 12. Complete solutions to Bessel’s equation may be written as c1 Jn (x) + c2 J−n (x)

if n is not an integer

or, for any value of n, c1 Jn (x) + c2 Yn (x)

c1 Hn(1) x + c2 Hn(2) (x)

or

13. The modified (or hyperbolic) Bessel’s differential equation is x2

dy d2 y +x − (x2 + n2 )y = 0 dx2 dx

14. When n is not an integer, two independent solutions of the equation are In (x) and I−n (x), where ∞  x n+2k  1 In (x) = k!Γ(n + k + 1) 2 k=0

15. If n is an integer, In (x) = I−n (x) =

xn 2n n!

"

x2 x4 + 22 · 1!(n + 1) 24 · 2!(n + 1)(n + 2) # x6 + 6 + ··· 2 · 3!(n + 1) (n + 2) (n + 3) 1+

16. For N = 0 and n = 1, this formula becomes I0 (x) = 1 + I1 (x) =

x 2

+

x2 22 (1!)2 3

x 23 ·1!2!

+

+

x4 24 (2!)2 5

x 25 ·2!3!

+

+

x6 26 (3!)2 7

x 27 ·3!4!

17. Another solution to the modified Bessel’s equation is $ 1 I−n (x)−In (x) 2π sin (nπ) Kn (x) = (x)−Iv (x) lim 21 π I−vsin (vπ) v→n

+

+ 9

x8 28 (4!)2

x 29 ·4!5!

+ ···

+ ···

n not an integer n an integer

This function is linearly independent of In (x) for all values of n. Thus the complete solution to the modified Bessel’s equation may be written as c1 In (x) + c2 I−n (x) n not an integer or c1 In (x) + c2 Kn (x)

any n

18. The following relations hold among the various Bessel functions: In (z) = i−m Jm (iz) Yn (iz) = (i)n+1 In (z) − π2 i−n Kn (z) Most of the properties of the modified Bessel function may be deduced from the known properties of Jn (x) by use of these relations and those previously given. 19. Recurrence formulas  In−1 (x) − In+1 (x) = 2n x In (x) In−1 (x) + In+1 (x) = 2In (x) n n   In (x) = In+1 (x) + x In (z) In−1 (x) − x In (x) = In (x)

A-95

TeamLRN

SPECIAL FUNCTIONS The Factorial Function For non-negative integers n, the factorial of n, denoted n!, is the product of all positive integers less than or equal to n; n! = n · (n − 1) · (n − 2) · · · 2 · 1. If n is a negative integer (n = −1, −2, . . . ) then n! = ±∞. Approximations to n! for large n include Stirling’s formula  n n+ 12 √ n! ≈ 2πe , e and Burnsides’s formula  n+ 12 √ n + 12 n! ≈ 2π . e n 0 2 4 6 8 10 12 14 16 18 20 30 50 70 90 110 130 500

n! 1 2 24 720 40320 3.6288 × 106 4.7900 × 108 8.7178 × 1010 2.0923 × 1013 6.4024 × 1015 2.4329 × 1018 2.6525 × 1032 3.0414 × 1064 1.1979 × 10100 1.4857 × 10138 1.5882 × 10178 6.4669 × 10219 1.2201 × 101134

Definition: Γ(n) =

∞

log10 n! 0.00000 0.30103 1.38021 2.85733 4.60552 6.55976 8.68034 10.94041 13.32062 15.80634 18.38612 32.42366 64.48307 100.07841 138.17194 178.20092 219.81069 1134.0864

n 1 3 5 7 9 11 13 15 17 19 25 40 60 80 100 120 150 1000

n! 1 6 120 5040 3.6288 × 105 3.9917 × 107 6.2270 × 109 1.3077 × 1012 3.5569 × 1014 1.2165 × 1017 1.5511 × 1025 8.1592 × 1047 8.3210 × 1081 7.1569 × 10118 9.3326 × 10157 6.6895 × 10198 5.7134 × 10262 4.0239 × 102567

log10 n! 0.00000 0.77815 2.07918 3.70243 5.55976 7.60116 9.79428 12.11650 14.55107 17.08509 25.19065 47.91165 81.92017 118.85473 157.97000 198.82539 262.75689 2567.6046

The Gamma Function t

n−1 −t

e

dt

n>0

0

Recursion Formula: Γ(n + 1) = nΓ(n) Γ(n + 1) = n! if n = 0, 1, 2, . . . where 0! = 1 For n < 0 the gamma function can be defined by using Γ(n) = Γ(n+1) n Graph:

A-96

SPECIAL FUNCTIONS Special Values:

Γ(1/2) =

√

π

1 · 3 · 5 · · · (2m − 1) √ π m = 1, 2, 3, . . . 2m √ (−1)m 2m π m = 1, 2, 3, . . . Γ(−m + 1/2) = 1 · 3 · 5 · · · (2m − 1)

Γ(m + 1/2) =

Definition:

1 · 2 · 3···k kx k→∞ (x + 1) (x + 2) · · · (x + k) ∞ %  −x/m &  x 1+ m e = xeγx

Γ(x + 1) = lim 1 Γ(x)

m=1

This is an infinite product representation for the gamma function where γ is Euler’s constant. Properties: ∞ Γ (1) = eγx ln x dx = −γ 0       Γ (x) 1 1 1 1 1 1 = −γ + − − − + + ... + + ··· Γ(x) 1 x " 2 x+1 n x + n#− 1 √ 1 139 1 Γ(x + 1) = 2πx xx e−x 1 + − + ... + 2 12x 288x 51, 840x3 This is called Stirling’s asymptotic series. ∞ Values of Γ(n) = 0 e−x xn−1 dx; n 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24

Γ(n) 1.00000 .99433 .98884 .98355 .97844 .97350 .96874 .96415 .95973 .95546 .95135 .94740 .94359 .93993 .93642 .93304 .92980 .92670 .92373 .92089 .91817 .91558 .91311 .91075 .90852

n 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

Γ(n) .90640 .90440 .90250 .90072 .89904 .89747 .89600 .89464 .89338 .89222 .89115 .89018 .88931 .88854 .88785 .88726 .88676 .88636 .88604 .88581 .88566 .88560 .88563 .88575 .88595

n 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74

A-97

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Γ(n + 1) = nΓ(n) Γ(n) .88623 .88659 .88704 .88757 .88818 .88887 .88964 .89049 .89142 .89243 .89352 .89468 .89592 .89724 .89864 .90012 .90167 .90330 .90500 .90678 .90864 .91057 .91258 .91466 .91683

n 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00

Γ(n) .91906 .92137 .92376 .92623 .92877 .93138 .93408 .93685 .93969 .94261 .94561 .94869 .95184 .95507 .95838 .96177 .96523 .96877 .97240 .97610 .97988 .98374 .98768 .99171 .99581 1.00000

SPECIAL FUNCTIONS Definition: B(m, n) =

The Beta Function

1

t

m−1

(1 − t)

m−1

dt

m > 0, n > 0

0

Γ(m)Γ(n) Γ(m + n) B(m, n) = B(n, m)  π/2 B(m, n) = 2 0 sin2m−1 θ cos2n−1 θ dθ  ∞ tm−1 B(m, n) = 0 (1+t)m+n dt  1 m−1 (1−t)n−1 B(m, n) = rn (r + 1)m 0 t (r+t) dt m+n

Relationship with Gamma function: B(m, n) = Properties:

The Error Function x 2 −t2 Definition: erf(x) = √ e dt π 0 3  x 1 x5 1 x7 2 x− + − + ··· Series: erf (x) = √ 3 2! 5 3! 7 π Property: erf(x) = − erf(−x)

  1 x erf √ 2 2 0 √ To evaluate erf(2.3), one proceeds as follows: For √x2 = 2.3, one finds x = (2.3) ( 2) = 3.25. In the normal probability function table (page A-104), one finds the entry 0.4994 opposite the value 3.25. Thus erf(2.3) = 2(0.4994) = 0.9988. x

Relationship with Normal Probability Function f (t) :

f (t) dt =

2 erfc(z) = 1 − erf(z) = √ π



∞

e−t dt 2

z

is known as the complementary error function. Orthogonal Polynomials I: Legendre Name: Legendre Symbol: Pn (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y  − 2 xy  + n(n + 1)y = 0 y = Pn (x)    [n/2] 1  n 2n − 2m n−2m Explicit Expression: Pn (x) = n (−1)m x 2 m=0 m n Recurrence Relation: (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x) Weight: 1 Standardization: Pn (1)=1 +1 2 [Pn (x)]2 dx = Norm: 2n +1 −1 (−1)n dn {(1 − x2 )n } Rodrigues’ Formula: Pn (x) = n 2 n! dxn ∞  Generating Function: R−1 = Pn (x)z n ; −1 < x < 1, |z| < 1, √ n=0 R = 1 − 2xz + z 2 Inequality: |Pn (x)| ≤ 1, −1 ≤ x ≤ 1. II: Tschebysheff, First Kind Name: Tschebysheff, First Kind Symbol: Tn (x) Interval:[-1, 1] Differential Equation: (1 − x2 )y − xy  + n2 y = 0 y = Tn (x) [n/2] n  (n − m − 1)! Explicit Expression: (2x)n−2m = cos(n arccos x) = Tn (x) (−1)m 2 m=0 m!(n − 2m)! A-98

SPECIAL FUNCTIONS Recurrence Relation: Tn+1 (x) = 2xTn (x) − Tn−1 (x) Weight: (1 − x2 )−1/2 Standardization: Tn (1) = 1 "  +1 π/2, n
= 0 Norm: −1 (1 − x2 )−1/2 [Tn (x)]2 dx = π, n = 0 √ (−1)n (1 − x2 )1/2 π dn {(1 − x2 )n−(1/2) } = Tn (x) Rodrigues’ Formula: dxn 2n+1 Γ(n + 12 ) ∞  1 − xz Generating Function: = Tn (x) z n , −1 < x < 1, |z| < 1 1 − 2xz − z 2 n=0 Inequality: |Tn (x)| ≤ 1, −1 ≤ x ≤ 1. III: Tschebysheff, Second Kind Name: Tschebysheff, Second Kind Symbol Un (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y  − 3 xy  + n(n + 2)y = 0 y = Un (x) [n/2]  (m − n)! Explicit Expression: Un (x) = (2x)n−2m (−1)m m!(n − 2m)! m=0 sin[(n + 1)θ] Un (cos θ) = sin θ Recurrence Relation: Un+1 (x) = 2xUn (x) − Un−1 (x) Standardization: Un (1) = n + 1 Weight: (1 − x2 )1/2 +1 π 2 1/2 2 Norm: (1 − x ) [Un (x)] dx = 2 −1 √ (−1)n (n + 1) π dn 2 n+(1/2) {(1 − x ) } Rodrigues’ Formula: Un (x) = (1 − x2 )1/2 2n+1 Γ(n + 32 ) dxn ∞  1 n Generating Function: = Un (x)z , − 1 < x < 1, |z| < 1 1 − 2xz + z 2 n=0 Inequality: |Un (x)| ≤ n + 1, −1 ≤ x ≤ 1.

IV: Jacobi (α,β) (x) Interval: [-1, 1] Name: Jacobi Symbol: Pn Differential Equation: (1 − x2 )y  + [β − α − (α + β + 2)x]y  + n(n + α + β + 1)y = 0 (α,β) y = Pn (x)    n  1 n + α n + β Explicit Expression: Pn(α,β) (x) = n (x − 1)n−m (x + 1)m 2 m=0 m n−m Recurrence Relation: (α,β)

2(n + 1) (n + α + β + 1) (2n + α + β)Pn+1 (x) = (2n + α + β + 1)[(α2 − β 2 ) + (2n + α + β + 2) × (2n + α + β)x]Pn(α,β) (x) (α,β)

− 2(n + α) (n + β) (2n + α + β + 2)Pn−1 (x)   (α,β) Weight: (1 − x)α (1 + x)β ; α, β > 1 Standardization: Pn (x) = n+α n +1 2α+β+1 Γ(n + α + 1)Γ(n + β + 1) Norm: (1 − x)α (1 + x)β [Pn(α,β) (x)]2 dx = (2n + α + β + 1)n!Γ(n + α + β + 1) −1 dn (−1)n (α,β) Rodrigues’ Formula: Pn (x) = n {(1 − x)n+α (1 + x)n+β } α β 2 n!(1 − x) (1 + x) dxn

A-99

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SPECIAL FUNCTIONS ∞  Generating Function: R−1 (1 − z + R)−α (1 + z + R)−β = 2−α−β Pn(α,β) (x)z n , n=0 √ R = 1 − 2xz+ z 2 ,  |z| < 1  n+q 1   ∼ nq if q = max(α, β) ≥ −   n 2  (α,β)  Inequality: max |Pn(α,β) (x)| = (x )| ∼ n−1/2 if q < − 12 |Pn −1≤x≤1   x is one of the two maximum points nearest    β−α α+β+1

V: Generalized Laguerre (α) Name: Generalized Laguerre Symbol: Ln (x) Interval: [0, ∞]   xy + (α + 1 − x)y + ny = 0 Differential Equation: (α) y = Ln (x)   n  1 m (α) m n+α (−1) x Explicit Expression: Ln (x) = n − m m! m=0 (α)

(α)

(α)

Recurrence Relation: (n + 1)Ln + 1(x) = [(2n + α + 1) − x]Ln (x) − (n + α)Ln − 1(x) n (α) Weight: xα e−x , α > −1 Standardization: Ln (x) = (−1) xn + · · · n! ∞ Γ(n + α + 1) 2 Norm: xα e−x [L(α) n (x)] dx = n! 0 dn 1 (α) {xn+α e−x } Rodrigues’ Formula: Ln (x) = n!xα e−xdxn∞ Generating Function: (1 − z)−α−1 exp

xz z−1

=



(α)

Ln (x)z n

n=0

Γ(n + α + 1) x/2 x≥0 e ; α>0 n!Γ(α + 1)   x≥0 (a) Γ(α+n+1) x/2 |Ln (x)| ≤ 2 − n!Γ(α+1) e ; −1 < α < 0

Inequality: |L(α) n (x) ≤

VI: Hermite Name: Hermite Symbol:Hn (x) Interval: [−∞, ∞] Differential Equation: y  − 2xy  + 2ny = 0 [n/2]  (−1)m n!(2x)n−2m Explicit Expression: Hn (x) = m!(n − 2m)! m=0 Recurrence Relation:Hn+1 (x) = 2xHn (x) − 2nHn−1 (x) 2 Weight: e−x Standardization: Hn (1) = 2n xn + · · · ∞ 2 √ 2 e−x [Hn (x)] dx = 2n n! π Norm: −∞

2

n

2

−x d ) Rodrigues’ Formula: Hn (x) = (−1)n ex dx n (e ∞ n  2 z Generating Function: e−x +2zx = Hn (x) n! n=0 √ x2 /2 n/2 k2 n! k ≈ 1.086435 Inequality: |Hn (x)|e

Tables of Orthogonal Polynomials

H0 = 1 H1 = 2x H2 = 4x2 − 2

x10 = (30240H0 + 75600H2 + 25200H4 + 2520H6 + 90H8 + H10 )/1024 x9 = (15120H1 + 10080H3 + 1512H5 + 72H7 + H9 )/512 x8 = (1680H0 + 3360H2 + 840H4 + 56H6 + H8 )/256

A-100

H3 = 8x3 − 12x x7 = (840H1 + 420H3 + 42H5 + H7 )/128 4 2 H4 = 16x − 48x + 12 x6 = (120H0 + 180H2 + 30H4 + H6 )/64 5 3 H5 = 32x − 160x + 120x x5 = (60H1 + 20H3 + H5 )/32 H6 = 64x6 − 480x4 + 720x2 − 120 x4 = (12H0 + 12H2 + H4 )/16 H7 = 128x7 − 1344x5 + 3360x3 − 1680x x3 = (6H1 + H3 )/8 8 6 4 2 H8 = 256x − 3584x + 13440x − 13440x + 1680 x2 = (2H0 + H2 )/4 9 7 5 3 H9 = 512x − 9216x + 48384x − 80640x + 30240x x = (H1 )/2 H10 = 1024x10 − 23040x8 + 161280x6 − 403200x4 + 302400x2 − 30240 1 = H0 L0 L1 L2 L3 L4 L5 L6

=1 x6 = 720L0 − 4320L1 + 10800L2 − 14400L3 + 10800L4 − 4320L5 + 720L6 = −x + 1 x5 = 120L0 − 600L1 + 1200L2 − 1200L3 + 600L4 − 120L5 = (x2 − 4x + 2)/2 x4 = 24L0 − 96L1 + 144L2 − 96L3 + 24L4 = (−x3 + 9x2 − 18x + 6)/6 x3 = 6L0 − 18L1 + 18L2 − 6L3 4 3 2 = (x − 16x + 72x − 96x + 24)/24 x2 = 2L0 − 4L1 + 2L2 5 4 3 2 = (−x + 25x − 200x + 600x − 600x + 120)/120 x = L0 − L1 = (x6 − 36x5 + 450x4 − 2400x3 + 5400x2 − 4320x + 720)/720 1 = L0

P0 = 1 x10 = (4199P0 + 16150P2 + 15504P4 + 7904P6 + 2176P8 + 256P10 )/46189 P1 = x x9 = (3315P1 + 4760P3 + 2992P5 + 960P7 + 128P9 )/12155 2 P2 = (3x − 1)/2 x8 = (715P0 + 2600P2 + 2160P4 + 832P6 + 128P8 )/6435 P3 = (5x3 − 3x)/2 x7 = (143P1 + 182P3 + 88P5 + 16P7 )/429 4 2 P4 = (35x − 30x + 3)/8 x6 = (33P0 + 110P2 + 72P4 + 16P6 )/231 5 3 P5 = (63x − 70x + 15x)/8 x5 = (27P1 + 28P3 + 8P5 )/63 6 4 2 P6 = (231x − 315x + 105x − 5)/16 x4 = (7P0 + 20P2 + 8P4 )/35 P7 = (429x7 − 693x5 + 315x3 − 35x)/16 x3 = (3P1 + 2P3 )/5 8 6 4 2 P8 = (6435x − 12012x + 6930x − 1260x + 35)/128 x2 = (P0 + 2P2 )/3 9 7 5 3 P9 = (12155x − 25740x + 18018x − 4620x + 315x)/128 x = P1 P10 = (46189x10 − 109395x8 + 90090x6 − 30030x4 + 3465x2 − 63)/256 1 = P0 T0 = 1 x10 = (126T0 + 210T2 + 120T4 + 45T6 + 10T8 + T10 )/512 T1 = x x9 = (126T1 + 84T3 + 36T5 + 9T7 + T9 )/256 2 T2 = 2x − 1 x8 = (35T0 + 56T2 + 28T4 + 8T6 + T8 )/128 3 T3 = 4x − 3x x7 = (35T1 + 21T3 + 7T5 + T7 )/64 T4 = 8x4 − 8x2 + 1 x6 = (10T0 + 15T2 + 6T4 + T6 )/32 T5 = 16x5 − 20x3 + 5x x5 = (10T1 + 5T3 + T5 )/16 6 4 2 T6 = 32x − 48x + 18x − 1 x4 = (3T0 + 4T2 + T4 )/8 T7 = 64x7 − 112x5 + 56x3 − 7x x3 = (3T1 + T3 )/4 T8 = 128x8 − 256x6 + 160x4 − 32x2 + 1 x2 = (T0 + T2 )/2 9 7 5 3 T9 = 256x − 576x + 432x − 120x + 9x x = T1 T10 = 512x10 − 1280x8 + 1120x6 − 400x4 + 50x2 − 1 1 = T0 U0 = 1 x10 = (42U0 + 90U2 + 75U4 + 35U6 + 9U8 + U10 )/1024 U1 = 2x x9 = (42U1 + 48U3 + 27U5 + 8U7 + U9 )/512 2 U2 = 4x − 1 x8 = (14U0 + 28U2 + 20U4 + 7U6 + U8 )/256 3 U3 = 8x − 4x x7 = (14U1 + 14U3 + 6U5 + U7 )/128 U4 = 16x4 − 12x2 + 1 x6 = (5U0 + 9U2 + 5U4 + U6 )/64 5 3 U5 = 32x − 32x + 6x x5 = (5U1 + 4U3 + U5 )/32 6 4 2 U6 = 64x − 80x + 24x − 1 x4 = (2U0 + 3U2 + U4 )/16 7 5 3 U7 = 128x − 192x + 80x − 8x x3 = (2U1 + U3 )/8 U8 = 256x8 − 448x6 + 240x4 − 40x2 + 1 x2 = (U0 + U2 )/4 9 7 5 3 U9 = 512x − 1024x + 672x − 160x + 10x x = (U1 )/2 U10 = 1024x10 − 2304x8 + 1792x6 − 560x4 + 60x2 − 1 1 = U0

A-101

TeamLRN

Clebsch–Gordan coefficients 

j1 m1

×

j2 j m2 m

 k

'

 = δm,m1 +m2

(j1 + j2 − j)!(j + j1 − j2 )!(j + j2 − j1 )!(2j + 1) (j + j1 + j2 + 1)!


(−1)k (j1 + m1 )!(j1 − m1 )!(j2 + m2 )!(j2 − m2 )!(j + m)!(j − m)! . k!(j1 + j2 − j − k)!(j1 − m1 − k)!(j2 + m2 − k)!(j − j2 + m1 + k)!(j − j1 − m2 + k)!

1. Conditions: (a) Each of {j1 , j2 , j, m1 , m2 , m} may be an integer, or half an integer. Additionally: j > 0, j1 > 0, j2 > 0 and j + j1 + j2 is an integer. (b) j1 + j2 − j ≥ 0. (c) j1 − j2 + j ≥ 0. (d) −j1 + j2 + j ≥ 0. (e) |m1 | ≤ j1 , |m2 | ≤ j2 , |m| ≤ j. 2. Special values:   j2 j j1 = 0 if m1 + m2 = m. (a) m m2 m   1 j1 0 j = δj1 ,j δm1 ,m . (b) m1 0 m   j 1 j2 j = 0 when j1 + j2 + j is an odd integer. (c) 0 0 0   j1 j j1 = 0 when 2j1 + j is an odd integer. (d) m1 m1 m   j1 j2 j : 3. Symmetry relations: all of the following are equal to m1 m2 m   j2 j1 j (a) , −m2 −m1 −m   j2 j1 j , (b) (−1)j1 +j2 −j m m2 m   1 j1 j2 j , (c) (−1)j1 +j2 −j −m1 −m2 −m    j j2 j1 2j+1 j2 +m2 (d) (−1) , 2j1 +1 −m m2 −m1    j j2 j1 2j+1 (−1)j1 −m1 +j−m , (e) 2j1 +1 m −m2 m1    j2 j j1 2j+1 , (−1)j−m+j1 −m1 (f) 2j1 +1 m2 −m −m1    j1 j j2 2j+1 (−1)j1 −m1 , (g) 2j2 +1 m1 −m −m2    j j1 j2 2j+1 (h) (−1)j1 −m1 . 2j2 +1 m −m1 m2 By use of the symmetry relations, Clebsch–Gordan coefficients may be put in the standard form j1 ≤ j2 ≤ j and m ≥ 0.

A-102

 m2

m

j1

j

− 12

0

1 2

1

0

1 2

1 2

1

1 2

0

1 2

1

1 2

1 2

1 2

1

1

1 2

1 2

j1 m1

√

1

m2

m

j1

j

−1

0

1

1

−1

0

1

2

− 12

0

1 2

3 2

− 12

1 2

1

1

− 12

1 2

1

2

0

0

1

2

0

0

1 2

3 2

0

1 2

1 2

3 2

0

1 2

1

1

0

1 2

1

2

0

1

1

1

2 2 √ 3 2 √ 2 2 √ 3 2

1 2 m2

j m



 m2

m

j1

j

≈ 0.707107

0

1

1

2

≈ 0.866025

1 2

0

1 2

3 2

≈ 0.707107

1 2

1 2

1

1

≈ 0.866025

1 2

1 2

1

2

1 2

1

1 2

3 2

1 2

3 2

1

2

1

0

1

1

√

1 ≈ 1.000000   j1 1 j m1 m2 m √ 2 2 √ 6 6 √ 2 2 3 4 √ 5 4 √ 6 3 √ 3 2 √ 6 3 √ 2 4 √ 10 4 √ 2 2

j1 m1

≈ 0.707107 ≈ 0.408248 ≈ 0.707107 ≈ 0.750000 ≈ 0.559017 ≈ 0.816496 ≈ 0.866025 ≈ 0.8164967 ≈ 0.353553 ≈ 0.790569 ≈ 0.707107

1

0

1

2

1

1 2

1 2

3 2

2 2 √ 2 2 √ − 42 √ 10 4 √ 30 6 √ 105 12 √ − 22 √ 6 6 √ 3 3

1

1 2

1

1

− 34

1

1 2

1

2

1

1

1 2

3 2

1

1

1

1

√

≈ 0.707107 ≈ 0.707107 ≈ −0.353553 ≈ 0.790569 ≈ 0.912871 ≈ 0.853913 ≈ −0.707107 ≈ 0.408248 ≈ 0.577350 ≈ −0.750000 ≈ 0.559017

10 4 √ − 22 √ 2 2

≈ 0.790569

1

≈ 1.000000

√

1

1

2

1

3 2

1 2

3 2

1

3 2

1

2

√ 105 12

1

2

1

2

1

TeamLRN



5 4

1

A-103

1 j m2 m

≈ −0.707107 ≈ 0.707107

≈ 0.853913 ≈ 1.000000

NORMAL PROBABILITY FUNCTION Table of the normal distribution For a standard normal random variable (Φ(z) is the area under the Standard Normal Curve from −∞ to z). Proportion of the total area (%) 68.27 90 95 95.45 99.0 99.73 99.8 99.9

Limits µ − λσ µ−σ µ − 1.65σ µ − 1.96σ µ − 2σ µ − 2.58σ µ − 3σ µ − 3.09σ µ − 3.29σ x Φ(x) 2[1 − Φ(x)] x 1 − Φ(x)

3.09 10−3

µ + λσ µ+σ µ + 1.65σ µ + 1.96σ µ + 2σ µ + 2.58σ µ + 3σ µ + 3.09σ µ + 3.29σ 1.282 0.90 0.20 3.72 10−4

1.645 0.95 0.10 4.26 10−5

1.960 0.975 0.05 4.75 10−6

Remaining area (%) 31.73 10 5 4.55 0.99 0.27 0.2 0.1

2.326 0.99 0.02 5.20 10−7

2.576 0.995 0.01 5.61 10−8

3.090 0.999 0.002 6.00 10−9

6.36 10−10

Areas under the Standard Normal Curve from 0 to z z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

0 .0000 .0398 .0793 .1179 .1554 .1915 .2258 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 4990 4993 4995 4997 4998 4998 4999 4999 5000

1 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2612 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .4991 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .5000

2 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2652 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .4991 .4994 .4995 .4997 .4998 .4999 .4999 .4999 .5000

3 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .4991 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

4 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2996 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

A-104

5 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

6 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

7 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .4992 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

8 .0319 .0714 .1103 .1480 .1844 .2190 .2518 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .4993 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

9 .0359 .0754 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .4999 .5000

Common sample size calculations Parameter

Estimate

Sample size

µ

x ¯

n=

p

pˆ

n=

µ2 − µ2

x ¯1 − x ¯2

n1 = n2 =

(zα/2 )2 (σ12 + σ22 ) E2

p1 − p2

pˆ1 − pˆ2

n1 = n2 =

(zα/2 )2 (p1 q1 + p2 q2 ) E2

z

α/2

· σ 2

E (zα/2 )2 · pq E2

Common one sample confidence intervals Parameter

Assumptions

100(1 − α)% Confidence interval

µ

n large, σ 2 known, or normality, σ 2 known

σ x ¯ ± zα/2 · √ n

µ

normality, σ 2 unknown

σ2

normality

p

binomial experiment, n large

s x ¯ ± tα/2,n−1 · √ n   (n − 1)s2 (n − 1)s2 , χ2α/2,n−1 χ21−α/2,n−1

pˆ(1 − pˆ) pˆ ± zα/2 · n

Common two sample confidence intervals 100(1 − α)% Confidence interval

Parameter

Assumptions

µ1 − µ2

normality, independence, σ12 , σ22 known or n1 , n2 large, independence, σ12 , σ22 known

(¯ x1 − x ¯2 ) ± zα/2 · (¯ x1 − x ¯2 ) ±

σ12

=

σ22

µ1 − µ2

normality, independence, unknown

µ1 − µ2

normality, independence, σ12 = σ22 unknown

σ2 σ12 + 2 n1 n2

1 1 + n1 n2 (n1 − 1)s21 + (n2 − 1)s22 s2p = n1 + n2 −2 s2 s21 (¯ x1 − x ¯2 ) ± tα/2,ν · + 2 n n 1 2  

t α2 ,n1 +n2 −2 · sp

ν≈

s2 1 n1

2 (s2 1 /n1 ) n1 −1

+ +

s2 2 n2

2

2 (s2 2 /n2 ) n2 −1

µ1 − µ2

normality, n pairs, dependence

sd d¯ ± tα/2,n−1 · √ n

p1 − p2

binomial experiments, n1 , n2 large, independence

(ˆ p1 − pˆ2 )±

pˆ2 (1 − pˆ2 ) pˆ1 (1 − pˆ1 ) zα/2 · + n1 n2

A-105

TeamLRN

PERCENTAGE POINTS, STUDENT’S t-DISTRIBUTION This table gives values of t such that F (t) =

    Γ n+1 x2 n+1 2  1 + dx − √ n n 2 nπΓ 2 −∞ for n, the number of degrees of freedom, equal to 1, 2, . . . , 30, 40, 60, 120, ∞; and for F (t) = 0.60, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, and 0.9995. The t-distribution is symmetrical, so that F (−t) = 1 − F (t) n/F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

.60 .325 .289 .277 .271 .267 .265 .263 .262 .261 .260 .260 .259 .259 .258 .258 .258 .257 .257 .257 .257 .257 .256 .256 .256 .256 .256 .256 .256 .256 .256 .255 .254 .254 .253

.75 1.000 .816 .765 .741 .727 .718 .711 .706 .703 .700 .697 .695 .694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .685 .684 .684 .684 .683 .683 .683 .681 .679 .677 .674

t

.90 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.296 1.289 1.282

.95 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 1.645

.975 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.980 1.960

.99 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.358 2.326

.995 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.617 2.576

.9995 636.619 31.598 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.767 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.373 3.291

*This table is abridged from the “Statistical Tables” of R. A. Fisher and Frank Yates published by Oliver & Boyd. Ltd., Edinburgh and London, 1938. It is here published with the kind permission of the authors and their publishers.

PERCENTAGE POINTS, CHI-SQUARE DISTRIBUTION This table gives values of χ2 such that

χ2

1   x(n−2)/2 e−x/2 dx n/2 Γ n 2 0 2 for n, the number of degrees equal to 1, 2, . . ., 30. For n > 30, a normal approximation is quite √ of freedom, √ accurate. The expression 2x2 − 2n − 1 is approximately normally distributed as the standard normal distribution. Thus χ2α , the α-point of the distribution, may be computed by the formula √ 1 χ2α = [xα + 2n − 1]2 , 2 F (χ)2 =

where xα is the α-point of the cumulative normal distribution. For even values of n, F (χ2 ) can be written as 1 − F (χ2 ) =

 x −1

x=0

A-106

e−λ λx x!

with λ = 12 χ2 and x = 12 n. Thus the cumulative Chi-Square distribution is related to the cumulative Poisson distribution. Another approximate formula for large n 

3 2 2 2 χα = n 1 − + zα 9n 9n n = degrees of freedom zα = the normal deviate (the value of x for which F (x) = the desired percentile). x 1.282 1.645 1.960 2.326 2.576 3.090 F (x) .90 .95 .975 .99 .995 .999 χ2.99 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 88.4 is the 99th percentile for 60 degrees of freedom.

χ2

1   xn−2/2 e−x/2 dx 2n/2 Γ n2

F (χ2 ) = 0

( n F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

.005 .0000393 .0100 .0717 .207 .412 .676 .989 1.34 1.73 2.16 2.60 3.07 3.57 4.07 4.60 5.14 5.70 6.26 6.84 7.43 8.03 8.64 9.26 9.89 10.5 11.2 11.8 12.5 13.1 13.8

.010 .000157 .0201 .115 .297 .554 .872 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54 10.2 10.9 11.5 12.2 12.9 12.6 14.3 15.0

.025 .000982 .0506 .216 .484 .831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.26 6.91 7.56 8.23 8.91 9.59 10.3 11.0 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8

.050 .00393 .103 .352 .711 1.15 1.64 2.17 2.73 3.33 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.1 10.9 11.6 12.3 13.1 13.8 14.6 15.4 16.2 16.9 17.7 18.5

.100 .0158 .211 .584 1.06 1.61 2.20 2.83 3.49 4.17 4.87 5.58 6.30 7.04 7.79 8.55 9.31 10.1 10.9 11.7 12.4 13.2 14.0 14.8 15.7 16.5 17.3 18.1 18.9 19.8 20.6

.250 .102 .575 1.21 1.92 2.67 3.45 4.25 5.07 5.90 6.74 7.58 8.44 9.30 10.2 11.0 11.9 12.8 13.7 14.6 15.5 16.3 17.2 18.1 19.0 19.9 20.8 21.7 22.7 23.6 24.5

.500 .455 1.39 2.37 3.36 4.35 5.35 6.35 7.34 8.34 9.34 10.3 11.3 12.3 13.3 14.3 15.3 16.3 17.3 18.3 19.3 20.3 21.3 22.3 23.3 24.3 25.3 26.3 27.3 28.3 29.3

.750 1.32 2.77 4.11 5.39 6.63 7.84 9.04 10.2 11.4 12.5 13.7 14.8 16.0 17.1 18.2 19.4 20.5 21.6 22.7 23.8 24.9 26.0 27.1 28.2 29.3 30.4 31.5 32.6 33.7 34.8

.900 2.71 4.61 6.25 7.78 9.24 10.6 12.0 13.4 14.7 16.0 17.3 18.5 19.8 21.1 22.3 23.5 24.8 26.0 27.2 28.4 29.6 30.8 32.0 33.2 34.4 35.6 36.7 37.9 39.1 40.3

.950 3.84 5.99 7.81 9.49 11.1 12.6 14.1 15.5 16.9 18.3 19.7 21.0 22.4 23.7 25.0 26.3 27.6 28.9 30.1 31.4 32.7 33.9 35.2 36.4 37.7 38.9 40.1 41.3 42.6 43.8

.975 5.02 7.38 9.35 11.1 12.8 14.4 16.0 17.5 19.0 20.5 21.9 23.3 24.7 26.1 27.5 28.8 30.2 31.5 32.9 34.2 35.5 36.8 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0

.990 6.63 9.21 11.3 13.3 15.1 16.8 18.5 20.1 21.7 23.2 24.7 26.2 27.7 29.1 30.6 32.0 33.4 34.8 36.2 37.6 38.9 40.3 41.6 43.0 44.3 45.6 47.0 48.3 49.6 50.9

PERCENTAGE POINTS, F -DISTRIBUTION This table gives values of F such that   F Γ m+n  m  2  n  mm/2 nn/2 xm−2/2 (n + mx)−(m+n)/2 dx F (F ) = Γ 2 Γ 2 0 for selected values of m, the number of degrees of freedom of the numerator of F ; and for selected values of n, the number of degrees freedom of the denominator of F . The table also provides values corresponding to F (F )=.10,.05,.025,.01,.005,.001 since F1−α for m and n degrees of freedom is the reciprocal of Fα for n and m degrees of freedom. Thus 1 1 = = .164 F.05 (4, 7) = F.95 (7, 4) 6.09

A-107

TeamLRN

.995 7.88 10.6 12.8 14.9 16.7 18.5 20.3 22.0 23.6 25.2 26.8 28.3 29.8 31.3 32.8 34.3 35.7 37.2 38.6 40.0 41.4 42.8 44.2 45.6 46.9 48.3 49.6 51.0 52.3 53.7



F

F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

  Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .90 Γ m Γ 2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

39.86 8.53 5.54 4.54 4.06 3.78 3.59 3.46 3.36 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.90 2.89 2.89 2.88 2.84 2.79 2.75 2.71

49.50 9.00 5.46 4.32 3.78 3.46 3.26 3.11 3.01 2.92 2.86 2.81 2.76 2.73 2.70 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.44 2.39 2.35 2.30

53.59 9.16 5.39 4.19 3.62 3.29 3.07 2.92 2.81 2.73 2.66 2.61 2.56 2.52 2.49 2.46 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.28 2.23 2.18 2.13 2.08

55.83 9.24 5.34 4.11 3.52 3.18 2.96 2.81 2.69 2.61 2.54 2.48 2.43 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.17 2.16 2.15 2.14 2.09 2.04 1.99 1.94

57.24 9.29 5.31 4.05 3.45 3.11 2.88 2.73 2.61 2.52 2.45 2.39 2.35 2.31 2.27 2.24 2.22 2.20 2.18 2.16 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.06 2.06 2.05 2.00 1.95 1.90 1.85

58.20 9.33 5.28 4.01 3.40 3.05 2.83 2.67 2.55 2.46 2.39 2.33 2.28 2.24 2.21 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02 2.01 2.00 2.00 1.99 1.98 1.93 1.87 1.82 1.77

58.91 9.35 5.27 3.98 3.37 3.01 2.78 2.62 2.51 2.41 2.34 2.28 2.23 2.19 2.16 2.13 2.10 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.87 1.82 1.77 1.72

59.44 9.37 5.25 3.95 3.34 2.98 2.75 2.59 2.47 2.38 2.30 2.24 2.20 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.83 1.77 1.72 1.67

59.86 9.38 5.24 3.94 3.32 2.96 2.72 2.56 2.44 2.35 2.27 2.21 2.16 2.12 2.09 2.06 2.03 2.00 1.98 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.87 1.86 1.85 1.79 1.74 1.68 1.63

60.19 9.39 5.23 3.92 3.30 2.94 2.70 2.54 2.42 2.32 2.25 2.19 2.14 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.88 1.87 1.86 1.85 1.84 1.83 1.82 1.76 1.71 1.65 1.60

60.71 9.41 5.22 3.90 3.27 2.90 2.67 2.50 2.38 2.28 2.21 2.15 2.10 2.05 2.02 1.99 1.96 1.93 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 1.71 1.66 1.60 1.55

61.22 9.42 5.20 3.87 3.24 2.87 2.63 2.46 2.34 2.24 2.17 2.10 2.05 2.01 1.97 1.94 1.91 1.89 1.86 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.66 1.60 1.55 1.49

61.74 9.44 5.18 3.84 3.21 2.84 2.59 2.42 2.30 2.20 2.12 2.06 2.01 1.96 1.92 1.89 1.86 1.84 1.81 1.79 1.78 1.76 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.61 1.54 1.48 1.42

62.00 9.45 5.18 3.83 3.19 2.82 2.58 2.40 2.28 2.18 2.10 2.04 1.98 1.94 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.72 1.70 1.69 1.68 1.67 1.66 1.65 1.64 1.57 1.51 1.45 1.38

62.26 9.46 5.17 3.82 3.17 2.80 2.56 2.38 2.25 2.16 2.08 2.01 1.96 1.91 1.87 1.84 1.81 1.78 1.76 1.74 1.72 1.70 1.69 1.67 1.66 1.65 1.64 1.63 1.62 1.61 1.54 1.48 1.41 1.34

62.53 9.47 5.16 3.80 3.16 2.78 2.54 2.36 2.23 2.13 2.05 1.99 1.93 1.89 1.85 1.81 1.78 1.75 1.73 1.71 1.69 1.67 1.66 1.64 1.63 1.61 1.60 1.59 1.58 1.57 1.51 1.44 1.37 1.30

62.79 9.47 5.15 3.79 3.14 2.76 2.51 2.34 2.21 2.11 2.03 1.96 1.90 1.86 1.82 1.78 1.75 1.72 1.70 1.68 1.66 1.64 1.62 1.61 1.59 1.58 1.57 1.56 1.55 1.54 1.47 1.40 1.32 1.24

63.06 9.48 5.14 3.78 3.12 2.74 2.49 2.32 2.18 2.08 2.00 1.93 1.88 1.83 1.79 1.75 1.72 1.69 1.67 1.64 1.62 1.60 1.59 1.57 1.56 1.54 1.53 1.52 1.51 1.50 1.42 1.35 1.26 1.17

63.33 9.49 5.13 3.76 3.10 2.72 2.47 2.29 2.16 2.06 1.97 1.90 1.85 1.80 1.76 1.72 1.69 1.66 1.63 1.61 1.59 1.57 1.55 1.53 1.52 1.50 1.49 1.48 1.47 1.46 1.38 1.29 1.19 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on mandndegrees of freedom, respectively.   F Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .95 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

161.4 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.00 3.92 3.84

199.5 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.15 3.07 3.00

215.7 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.76 2.68 2.60

224.6 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.53 2.45 2.37

230.2 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.37 2.29 2.21

234.0 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.25 2.17 2.10

236.8 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.17 2.09 2.01

238.9 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.10 2.02. 1.94

240.5 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.04 1.96 1.88

241.9 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 1.99 1.91 1.83

243.9 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 1.92 1.83 1.75

245.9 19.43 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.84 1.75 1.67

248.0 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.84 1.75 1.66 1.57

249.1 19.45 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.95 1.93 1.91 1.90 1.89 1.79 1.70 1.61 1.52

250.1 19.46 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.90 1.88 1.87 1.85 1.84 1.74 1.65 1.55 1.46

251.1 19.47 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.85 1.84 1.82 1.81 1.79 1.69 1.59 1.50 1.39

252.2 19.48 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.79 1.77 1.75 1.74 1.64 1.53 1.43 1.32

253.3 19.49 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.71 1.70 1.68 1.58 1.47 1.35 1.22

254.3 19.50 8.53 5.63 4.36 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 139 1.25 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on m and n degrees of freedom, respectively.

A-108



F

F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

  Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .975 Γ m Γ 2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

647.8 38.51 17.44 12.22 10.01 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.41 6.30 6.20 6.12 6.04 5.98 5.92 5.87 5.83 5.79 5.75 5.72 5.69 5.66 5.63 5.61 5.59 5.57 5.42 5.29 5.15 5.02

799.5 39.00 16.04 10.65 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10 4.97 4.86 4.77 4.69 4.62 4.56 4.51 4.46 4.42 4.38 4.35 4.32 4.29 4.27 4.24 4.22 4.20 4.18 4.05 3.93 3.80 3.69

864.2 39.17 15.44 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.01 3.95 3.90 3.86 3.82 3.78 3.75 3.72 3.69 3.67 3.65 3.63 3.61 3.59 3.46 3.34 3.23 3.12

899.6 39.25 15.10 9.60 7.39 6.23 5.52 5.05 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56 3.51 3.48 3.44 3.41 3.38 3.35 3.33 3.31 3.29 3.27 3.25 3.13 3.01 2.89 2.79

921.8 39.30 14.88 9.36 7.15 5.99 5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58 3.50 3.44 3.38 3.33 3.29 3.25 3.22 3.18 3.15 3.13 3.10 3.08 3.06 3.04 3.03 2.90 2.79 2.67 2.57

937.1 39.33 14.73 9.20 6.98 5.82 5.12 4.65 4.32 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17 3.13 3.09 3.05 3.02 2.99 2.97 2.94 2.92 2.90 2.88 2.87 2.74 2.63 2.52 2.41

948.2 39.36 14.62 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05 3.01 2.97 2.93 2.90 2.87 2.85 2.82 2.80 2.78 2.76 2.75 2.62 2.51 2.39 2.29

956.7 39.37 14.54 8.98 6.76 5.60 4.90 4.43 4.10 3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01 2.96 2.91 2.87 2.84 2.81 2.78 2.75 2.73 2.71 2.69 2.67 2.65 2.53 2.41 2.30 2.19

963.3 39.39 14.47 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88 2.84 2.80 2.76 2.73 2.70 2.68 2.65 2.63 2.61 2.59 2.57 2.45 2.33 2.22 2.11

968.6 39.40 14.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82 2.77 2.73 2.70 2.67 2.64 2.61 2.59 2.57 2.55 2.53 2.51 2.39 2.27 2.16 2.05

976.7 39.41 14.34 8.75 6.52 5.37 4.67 4.20 3.87 3.62 3.43 3.28 3.15 3.05 2.96 2.89 2.82 2.77 2.72 2.68 2.64 2.60 2.57 2.54 2.51 2.49 2.47 2.45 2.43 2.41 2.29 2.17 2.05 1.94

984.9 39.43 14.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3.33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62 2.57 2.53 2.50 2.47 2.44 2.41 2.39 2.36 2.34 2.32 2.31 2.18 2.06 1.94 1.83

993.1 39.45 14.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.68 2.62 2.56 2.51 2.46 2.42 2.39 2.36 2.33 2.30 2.28 2.25 2.23 2.21 2.20 2.07 1.94 1.82 1.71

997.2 39.46 14.12 8.51 6.28 5.12 4.42 3.95 3.61 3.37 3.17 3.02 2.89 2.79 2.70 2.63 2.56 2.50 2.45 2.41 2.37 2.33 2.30 2.27 2.24 2.22 2.19 2.17 2.15 2.14 2.01 1.88 1.76 1.64

1001 39.46 14.08 8.46 6.23 5.07 4.36 3.89 3.56 3.31 3.12 2.96 2.84 2.73 2.64 2.57 2.50 2.44 2.39 2.35 2.31 2.27 2.24 2.21 2.18 2.16 2.13 2.11 2.09 2.07 1.94 1.82 1.69 1.57

1006 39.47 14.04 8.41 6.18 5.01 4.31 3.84 3.51 3.26 3.06 2.91 2.78 2.67 2.59 2.51 2.44 2.38 2.33 2.29 2.25 2.21 2.18 2.15 2.12 2.09 2.03 2.05 2.03 2.01 1.88 1.74 1.61 1.48

1010 39.48 13.99 8.36 6.12 4.96 4.25 3.78 3.45 3.20 3.00 2.85 2.72 2.61 2.52 2.45 2.38 2.32 2.27 2.22 2.18 2.14 2.11 2.08 2.05 2.03 2.00 1.98 1.96 1.94 1.80 1.67 1.53 1.39

1014 39.49 13.95 8.31 6.07 4.90 4.20 3.73 3.39 3.14 2.94 2.79 2.66 2.55 2.46 2.38 2.32 2.26 2.20 2.16 2.11 2.08 2.04 2.01 1.98 1.95 1.93 1.91 1.89 1.87 1.72 1.58 1.43 1.27

1018 39.50 13.90 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.13 2.09 2.04 2.00 1.97 1.94 1.91 1.88 1.85 1.83 1.81 1.79 1.64 1.48 1.31 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on mandndegrees of freedom, respectively.   F Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .99 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

4052 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7.88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.08 6.85 6.63

4999.5 99.00 30.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 4.98 4.79 4.61

5403 99.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.13 3.95 3.78

5625 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.65 3.48 3.32

5764 99.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.85 3.82 3.78 3.75 3.73 3.70 3.51 3.34 3.17 3.02

5859 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.12 2.96 2.80

5928 99.36 27.67 14.98 10.46 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.42 3.39 3.36 3.33 3.30 3.12 2.95 2.79 2.64

5982 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.82 2.66 2.51

6022 99.39 27.35 14.66 10.16 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3.18 3.15 3.12 3.09 3.07 2.89 2.72 2.56 2.41

6056 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.31 3.26 3.21 3.17 3.13 3.09 3.06 3.03 3.00 2.98 2.80 2.63 2.47 2.32

6106 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.50 2.34 2.18

6157 99.43 26.87 14.20 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.81 2.78 2.75 2.73 2.70 2.52 2.35 2.19 2.04

6209 99.45 26.69 14.02 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.88 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.37 2.20 2.03 1.88

6235 99.46 26.60 13.93 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.80 2.75 2.70 2.66 2.62 2.58 2.55 2.52 2.49 2.47 2.29 2.12 1.95 1.79

6261 99.47 26.50 13.84 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.54 2.50 2.47 2.44 2.41 2.39 2.20 2.03 1.86 1.70

6287 99.47 26.41 13.75 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.30 2.11 1.94 1.76 1.59

6313 99.48 26.32 13.65 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2.55 2.50 2.45 2.40 2.36 2.33 2.29 2.26 2.23 2.21 2.02 1.84 1.66 1.47

6339 99.49 26.22 13.56 9.11 6.97 5.74 4.95 4.40 4.00 3.69 3.45 3.25 3.09 2.96 2.84 2.75 2.66 2.58 2.52 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.17 2.14 2.11 1.92 1.73 1.53 1.32

6366 99.50 26.13 13.46 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.06 2.03 2.01 1.80 1.60 1.38 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on m and n degrees of freedom, respectively.

A-109

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SPECIAL FUNCTIONS Bessel Functions 1. Bessel’s differential equation for a real variable x is x2

d2 y dy +x + (x2 − n2 )y = 0 2 dx dx

2. When n is not an integer, two independent solutions of the equation are Jn (x), J−n (x), where Jn (x) =

∞  k=0

x (−1)k k!Γ(n + k + 1) 2

n+2k

3. If n is an integer Jn (x) = (−1)n Jn (x), where # " xn x4 x6 x2 Jn (x) = n + 4 + 6 + ... 1− 2 2 n! 2 · 1!(n + 1) 2 · 2!(n + 1) (n + 2) 2 · 3!(n + 1) (n + 2) (n + 3) 4. For n = 0 and n = 1, this formula becomes x2 22 (1!)2 x x3 2 − 23 ·1!2!

J0 (x) = 1 −

+

J1 (x) =

+

x4 24 (2!)2 x5 25 ·2!3!

− −

x6 x8 26 (3!)2 + 28 (4!)2 − · · · x7 x9 27 ·3!4! + 29 ·4!5! − . . .

5. When x is large and positive, the following asymptotic series may be used 1   π π ! 2 2 − Q0 (x) sin x − P0 (x) cos x − πx 4 4  &  2  12 %   3π 3π J1 (x) = πx P1 (x) cos x − 4 − Q1 (x) sin x − 4 

J0 (x) =

where 12 · 32 12 · 32 · 52 · 72 12 · 32 · 52 · 72 · 92 · 112 + − + ··· 2!(8x)2 4!(8x)4 6!(8x)6 12 12 ·32 ·52 12 ·32 ·52 ·72 ·92 Q0 (x) ∼ − 1!8x + 3!(8x)3 − + −··· 5!(8x)5

P0 (x) ∼ 1 −

2 2 2 2 2 12 ·3·5 12 ·32 ·52 ·7·9 ·7 ·9 ·11·13 + 1 ·3 ·56!(8x) 6 2!(8x)2 − 4!(8x)4 2 2 2 2 2 2 1·3 1 ·3 ·5·7 1 ·3 ·5 ·7 ·9·11 − + − · · · 3 5 1!8x 3!(8x) 5!(8x)

P1 (x) ∼ 1 + Q1 (x) ∼

− +···

[In P1 (x) the signs alternate from+to-after the first term] 6. The zeros of J0 (x) and J1 (x). If j0s and j1s are the sth zeros of J0 (x) and J1 (x) respectively, and if a = 4s − 1, b = 4s + 1 # " 1 62 15, 116 12, 554, 474 8, 368, 654, 292 2 j0,s ∼ πa 1 + 2 2 − 4 4 + − + − +··· 10 a10 4 % π a 3π a 15π 6 a6 105π 8 a8 315π & 3,902,418 895,167,324 4716 j1,s ∼ 14 πb 1 − π26b2 + π46b4 − 5π 6 b6 + 35π 8 b8 − 35π 10 b10 + · · · & % s+1 3 2 (−1) 2 56 9664 J1 (j0,s ) ∼ 1 − 3π4 a4 + 5π6 a6 − 7,381,280 1 21π 8 a8 + · · · πa 23 & s 2 % 19,584 2,466,720 J0 (j1,s ) ∼ (−1) 12 1 + π24 4 b4 − 10π 6 b6 + 7π 8 b8 − · · · πb 2

A-93

SPECIAL FUNCTIONS 7. Table of zeros for J0 (x) and J1 (x) Define {αn , βn } by J1 (αn ) = 0 and J0 (βn ) = 0. Roots αn 2.4048 5.5201 8.6537 11.7915 14.9309 18.0711 21.2116

J1 (αn ) 0.5191 -0.3403 0.2715 -0.2325 0.2065 -0.1877 0.1733

Roots β n 0.0000 3.8317 7.0156 10.1735 13.3237 16.4706 19.6159

J0 (βn ) 1.0000 -0.4028 0.3001 -0.2497 0.2184 -0.1965 0.1801

8. Recurrence formulas 2n Jn (x) nJn (x) + xJn (x) = xJn−1 (x) x  Jn−1 (x) − Jn+1 (x) = 2Jn (x) nJn (x) − xJn (x) = xJn+1 (x)

Jn−1 (x) + Jn+1 (x) =

(k)

9. If Jn is written for Jn (x) and Jn relationships are important (r)

J0 (2) J0 (3) J0 (4) J0

is written for

(r−1)



J 12 (x) =

2

πx 

J− 12 (x) = Jn+ 32 (x) = Jn− 12 (x) =

sin x

2 πx cos x 1 d 1 −xn+ 2 dx {x−(n+ 2 ) Jn+ 12 (x)} 1 1 d x−(n+ 2 ) dx {xn+ 2 Jn+ 12 (x)}

 πx  12

n 0

2

x3

2

− cos x

 − 1 sin x − x3 cos x    − x6 sin x − x152 − 1 cos x etc.

  15

 πx  12

Jn+ 12 (x) sin x

sin x x

1 2

then the following derivative

= −J1 = −J0 + x1 J1 = 12 (J2 − J0 )   = x1 J0 + 1 − x22 J1 = 14 (−J3 + 3J1 )     = 1 − x32 J0 − x2 − x63 J1 = 18 (J4 − 4J2 + 3J0 ), etc.

10. Half order Bessel functions

3

dk {Jn (x)}, dxk



3 x2

−

 15 x3

J−(n+ 12 ) (x) cos x

3 x2

− cosx x − sin x  − 1 cos x + x3 sin x

−

6 x



cos x −

 15 x2

 − 1 sin x

11. Additional solutions to Bessel’s equation are Yn (x) (also called Weber’s function, and sometimes denoted by Nn (x)) (1) (2) Hn (x) and Hn (x) (also called Hankel functions) These solutions are defined as follows    Jn (x) cos (nπ) − J−n (x) sin(nπ) Yn (x) =  −v (x)  lim Jv (x) cos(vπ)−J sin(vπ) v→n

n

not an integer n

an integer

A-94

TeamLRN

(1)

Hn (x) = Jn (x) + iYn (x) (2) Hn (x) = Jn (x) − iYn (x)

SPECIAL FUNCTIONS The additional properties of these functions may all be derived from the above relations and the known properties of Jn (x). 12. Complete solutions to Bessel’s equation may be written as c1 Jn (x) + c2 J−n (x)

if n is not an integer

or, for any value of n, c1 Jn (x) + c2 Yn (x)

c1 Hn(1) x + c2 Hn(2) (x)

or

13. The modified (or hyperbolic) Bessel’s differential equation is x2

d2 y dy +x − (x2 + n2 )y = 0 dx2 dx

14. When n is not an integer, two independent solutions of the equation are In (x) and I−n (x), where ∞  x n+2k  1 In (x) = k!Γ(n + k + 1) 2 k=0

15. If n is an integer, In (x) = I−n (x) =

xn 2n n!

"

x2 x4 + 22 · 1!(n + 1) 24 · 2!(n + 1)(n + 2) # x6 + 6 + ··· 2 · 3!(n + 1) (n + 2) (n + 3) 1+

16. For N = 0 and n = 1, this formula becomes I0 (x) = 1 + I1 (x) =

x 2

+

x2 22 (1!)2 3

x 23 ·1!2!

+

+

x4 24 (2!)2 5

x 25 ·2!3!

+

+

x6 26 (3!)2 7

x 27 ·3!4!

17. Another solution to the modified Bessel’s equation is $ 1 I−n (x)−In (x) 2π sin (nπ) Kn (x) = (x)−Iv (x) lim 21 π I−vsin (vπ) v→n

+

+ 9

x8 28 (4!)2

x 29 ·4!5!

+ ···

+ ···

n not an integer n an integer

This function is linearly independent of In (x) for all values of n. Thus the complete solution to the modified Bessel’s equation may be written as c1 In (x) + c2 I−n (x) n not an integer or c1 In (x) + c2 Kn (x)

any n

18. The following relations hold among the various Bessel functions: In (z) = i−m Jm (iz) Yn (iz) = (i)n+1 In (z) − π2 i−n Kn (z) Most of the properties of the modified Bessel function may be deduced from the known properties of Jn (x) by use of these relations and those previously given. 19. Recurrence formulas  In−1 (x) − In+1 (x) = 2n x In (x) In−1 (x) + In+1 (x) = 2In (x) n n   In (x) = In+1 (x) + x In (z) In−1 (x) − x In (x) = In (x)

A-95

SPECIAL FUNCTIONS The Factorial Function For non-negative integers n, the factorial of n, denoted n!, is the product of all positive integers less than or equal to n; n! = n · (n − 1) · (n − 2) · · · 2 · 1. If n is a negative integer (n = −1, −2, . . . ) then n! = ±∞. Approximations to n! for large n include Stirling’s formula  n n+ 12 √ n! ≈ 2πe , e and Burnsides’s formula  n+ 12 √ n + 12 n! ≈ 2π . e n 0 2 4 6 8 10 12 14 16 18 20 30 50 70 90 110 130 500

n! 1 2 24 720 40320 3.6288 × 106 4.7900 × 108 8.7178 × 1010 2.0923 × 1013 6.4024 × 1015 2.4329 × 1018 2.6525 × 1032 3.0414 × 1064 1.1979 × 10100 1.4857 × 10138 1.5882 × 10178 6.4669 × 10219 1.2201 × 101134

log10 n! 0.00000 0.30103 1.38021 2.85733 4.60552 6.55976 8.68034 10.94041 13.32062 15.80634 18.38612 32.42366 64.48307 100.07841 138.17194 178.20092 219.81069 1134.0864

n 1 3 5 7 9 11 13 15 17 19 25 40 60 80 100 120 150 1000

A-96

TeamLRN

n! 1 6 120 5040 3.6288 × 105 3.9917 × 107 6.2270 × 109 1.3077 × 1012 3.5569 × 1014 1.2165 × 1017 1.5511 × 1025 8.1592 × 1047 8.3210 × 1081 7.1569 × 10118 9.3326 × 10157 6.6895 × 10198 5.7134 × 10262 4.0239 × 102567

log10 n! 0.00000 0.77815 2.07918 3.70243 5.55976 7.60116 9.79428 12.11650 14.55107 17.08509 25.19065 47.91165 81.92017 118.85473 157.97000 198.82539 262.75689 2567.6046

Definition: Γ(n) =

∞

The Gamma Function t

n−1 −t

e

dt

n>0

0

Recursion Formula: Γ(n + 1) = nΓ(n) Γ(n + 1) = n! if n = 0, 1, 2, . . . where 0! = 1 For n < 0 the gamma function can be defined by using Γ(n) = Γ(n+1) n Graph:

A-96

SPECIAL FUNCTIONS Special Values:

Γ(1/2) =

√

π

1 · 3 · 5 · · · (2m − 1) √ π m = 1, 2, 3, . . . 2m √ (−1)m 2m π m = 1, 2, 3, . . . Γ(−m + 1/2) = 1 · 3 · 5 · · · (2m − 1)

Γ(m + 1/2) =

Definition:

1 · 2 · 3···k kx k→∞ (x + 1) (x + 2) · · · (x + k) ∞ %  −x/m &  x 1+ m e = xeγx

Γ(x + 1) = lim 1 Γ(x)

m=1

This is an infinite product representation for the gamma function where γ is Euler’s constant. Properties: ∞ Γ (1) = eγx ln x dx = −γ 0       Γ (x) 1 1 1 1 1 1 = −γ + − − − + + ... + + ··· Γ(x) 1 x " 2 x+1 n x + n#− 1 √ 1 139 1 − + ... Γ(x + 1) = 2πx xx e−x 1 + + 2 12x 288x 51, 840x3 This is called Stirling’s asymptotic series. ∞ Values of Γ(n) = 0 e−x xn−1 dx; n 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24

Γ(n) 1.00000 .99433 .98884 .98355 .97844 .97350 .96874 .96415 .95973 .95546 .95135 .94740 .94359 .93993 .93642 .93304 .92980 .92670 .92373 .92089 .91817 .91558 .91311 .91075 .90852

n 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

Γ(n) .90640 .90440 .90250 .90072 .89904 .89747 .89600 .89464 .89338 .89222 .89115 .89018 .88931 .88854 .88785 .88726 .88676 .88636 .88604 .88581 .88566 .88560 .88563 .88575 .88595

n 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74

A-97

TeamLRN

Γ(n + 1) = nΓ(n) Γ(n) .88623 .88659 .88704 .88757 .88818 .88887 .88964 .89049 .89142 .89243 .89352 .89468 .89592 .89724 .89864 .90012 .90167 .90330 .90500 .90678 .90864 .91057 .91258 .91466 .91683

n 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00

Γ(n) .91906 .92137 .92376 .92623 .92877 .93138 .93408 .93685 .93969 .94261 .94561 .94869 .95184 .95507 .95838 .96177 .96523 .96877 .97240 .97610 .97988 .98374 .98768 .99171 .99581 1.00000

SPECIAL FUNCTIONS

1

The Beta Function tm−1 (1 − t)m−1 dt

Definition: B(m, n) =

m > 0, n > 0

0

Γ(m)Γ(n) Γ(m + n) B(m, n) = B(n, m)  π/2 B(m, n) = 2 0 sin2m−1 θ cos2n−1 θ dθ  ∞ tm−1 B(m, n) = 0 (1+t) m+n dt  1 m−1 (1−t)n−1 n B(m, n) = r (r + 1)m 0 t (r+t) dt m+n

Relationship with Gamma function: B(m, n) = Properties:

2 Definition: erf(x) = √

m!(n − 2m)!

(2x)n−2m = cos(n arccos x) = Tn (x)

A-98

(r+t)m+n

The Error Function x 2 −t2 Definition: erf(x) = √ e dt π 0 3  x 1 x5 1 x7 2 x− + − + ··· Series: erf (x) = √ 3 2! 5 3! 7 π Property: erf(x) = − erf(−x)

dt

  1 x erf √ 2 2 0 √ To evaluate erf(2.3), one proceeds as follows: For √x2 = 2.3, one finds x = (2.3) ( 2) = 3.25. In the normal probability function table (page A-104), one finds the entry 0.4994 opposite the value 3.25. Thus erf(2.3) = 2(0.4994) = 0.9988. x

Relationship with Normal Probability Function f (t) :

f (t) dt =

2 erfc(z) = 1 − erf(z) = √ π



∞

z

is known as the complementary error function. I: Legendre Name: Legendre Symbol: Pn (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y  − 2 xy  + n(n + 1)y = 0 y = Pn (x) Explicit Expression: Pn (x) =

1

TeamLRN

e−t dt 2

π

∞

e−t dt 2

z

Orthogonal Polynomials I: Legendre Name: Legendre Symbol: Pn (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y  − 2 xy  + n(n + 1)y = 0 y = Pn (x)    [n/2] 1  2n − 2m n−2m m n Explicit Expression: Pn (x) = n (−1) x 2 m=0 m n Recurrence Relation: (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x) Weight: 1 Standardization: Pn (1)=1 +1 2 Norm: [Pn (x)]2 dx = 2n + 1 −1 (−1)n dn {(1 − x2 )n } Rodrigues’ Formula: Pn (x) = n 2 n! dxn ∞  Generating Function: R−1 = Pn (x)z n ; −1 < x < 1, |z| < 1, √ n=0 R = 1 − 2xz + z 2 Inequality: |Pn (x)| ≤ 1, −1 ≤ x ≤ 1. II: Tschebysheff, First Kind Name: Tschebysheff, First Kind Symbol: Tn (x) Interval:[-1, 1] Differential Equation: (1 − x2 )y − xy  + n2 y = 0 y = Tn (x) [n/2] n  (n − m − 1)! Explicit Expression: (2x)n−2m = cos(n arccos x) = Tn (x) (−1)m 2 m=0 m!(n − 2m)!

A-98

SPECIAL FUNCTIONS Recurrence Relation: Tn+1 (x) = 2xTn (x) − Tn−1 (x) Weight: (1 − x2 )−1/2 Standardization: Tn (1) = 1 "  +1 π/2, n
= 0 Norm: −1 (1 − x2 )−1/2 [Tn (x)]2 dx = π, n = 0 √ (−1)n (1 − x2 )1/2 π dn {(1 − x2 )n−(1/2) } = Tn (x) Rodrigues’ Formula: dxn 2n+1 Γ(n + 12 ) ∞  1 − xz Generating Function: = Tn (x) z n , −1 < x < 1, |z| < 1 1 − 2xz − z 2 n=0 Inequality: |Tn (x)| ≤ 1, −1 ≤ x ≤ 1. III: Tschebysheff, Second Kind Name: Tschebysheff, Second Kind Symbol Un (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y  − 3 xy  + n(n + 2)y = 0 y = Un (x) [n/2]  (m − n)! Explicit Expression: Un (x) = (2x)n−2m (−1)m m!(n − 2m)! m=0 sin[(n + 1)θ] Un (cos θ) = sin θ Recurrence Relation: Un+1 (x) = 2xUn (x) − Un−1 (x) Standardization: Un (1) = n + 1 Weight: (1 − x2 )1/2 +1 π 2 1/2 2 Norm: (1 − x ) [Un (x)] dx = 2 −1 √ (−1)n (n + 1) π dn 2 n+(1/2) {(1 − x ) } Rodrigues’ Formula: Un (x) = (1 − x2 )1/2 2n+1 Γ(n + 32 ) dxn ∞  1 n Generating Function: = Un (x)z , − 1 < x < 1, |z| < 1 1 − 2xz + z 2 n=0 Inequality: |Un (x)| ≤ n + 1, −1 ≤ x ≤ 1.

IV: Jacobi (α,β) (x) Interval: [-1, 1] Name: Jacobi Symbol: Pn Differential Equation: (1 − x2 )y  + [β − α − (α + β + 2)x]y  + n(n + α + β + 1)y = 0 (α,β) y = Pn (x)    n  1 n + α n + β Explicit Expression: Pn(α,β) (x) = n (x − 1)n−m (x + 1)m 2 m=0 m n−m Recurrence Relation: (α,β)

2(n + 1) (n + α + β + 1) (2n + α + β)Pn+1 (x) = (2n + α + β + 1)[(α2 − β 2 ) + (2n + α + β + 2) × (2n + α + β)x]Pn(α,β) (x) (α,β)

− 2(n + α) (n + β) (2n + α + β + 2)Pn−1 (x)   (α,β) Weight: (1 − x)α (1 + x)β ; α, β > 1 Standardization: Pn (x) = n+α n +1 2α+β+1 Γ(n + α + 1)Γ(n + β + 1) Norm: (1 − x)α (1 + x)β [Pn(α,β) (x)]2 dx = (2n + α + β + 1)n!Γ(n + α + β + 1) −1 dn (−1)n (α,β) Rodrigues’ Formula: Pn (x) = n {(1 − x)n+α (1 + x)n+β } α β 2 n!(1 − x) (1 + x) dxn

A-99

TeamLRN

SPECIAL FUNCTIONS ∞  Generating Function: R−1 (1 − z + R)−α (1 + z + R)−β = 2−α−β Pn(α,β) (x)z n , n=0 √ R = 1 − 2xz+ z 2 ,  |z| < 1  n+q 1   ∼ nq if q = max(α, β) ≥ −   n 2  (α,β)  Inequality: max |Pn(α,β) (x)| = (x )| ∼ n−1/2 if q < − 12 |Pn −1≤x≤1   x is one of the two maximum points nearest    β−α α+β+1

V: Generalized Laguerre (α) Name: Generalized Laguerre Symbol: Ln (x) Interval: [0, ∞]   xy + (α + 1 − x)y + ny = 0 Differential Equation: (α) y = Ln (x)   n  1 m (α) m n+α (−1) x Explicit Expression: Ln (x) = n − m m! m=0 (α)

(α)

(α)

Recurrence Relation: (n + 1)Ln + 1(x) = [(2n + α + 1) − x]Ln (x) − (n + α)Ln − 1(x) n (α) Weight: xα e−x , α > −1 Standardization: Ln (x) = (−1) xn + · · · n! ∞ Γ(n + α + 1) 2 Norm: xα e−x [L(α) n (x)] dx = n! 0 dn 1 (α) {xn+α e−x } Rodrigues’ Formula: Ln (x) = n!xα e−xdxn∞ Generating Function: (1 − z)−α−1 exp

xz z−1

=



(α)

Ln (x)z n

n=0

Γ(n + α + 1) x/2 x≥0 e ; α>0 n!Γ(α + 1)   x≥0 (a) Γ(α+n+1) x/2 |Ln (x)| ≤ 2 − n!Γ(α+1) e ; −1 < α < 0

Inequality: |L(α) n (x) ≤

VI: Hermite Name: Hermite Symbol:Hn (x) Interval: [−∞, ∞] Differential Equation: y  − 2xy  + 2ny = 0 [n/2]  (−1)m n!(2x)n−2m Explicit Expression: Hn (x) = m!(n − 2m)! m=0 Recurrence Relation:Hn+1 (x) = 2xHn (x) − 2nHn−1 (x) 2 Weight: e−x Standardization: Hn (1) = 2n xn + · · · ∞ 2 √ 2 e−x [Hn (x)] dx = 2n n! π Norm: −∞

2

n

2

−x d ) Rodrigues’ Formula: Hn (x) = (−1)n ex dx n (e ∞ n  2 z Generating Function: e−x +2zx = Hn (x) n! n=0 √ x2 /2 n/2 k2 n! k ≈ 1.086435 Inequality: |Hn (x)|e

H0 = 1 H1 = 2x H2 = 4x2 − 2

x10 = (30240H0 + 75600H2 + 25200H4 + 2520H6 + 90H8 + H10 )/1024 x9 = (15120H1 + 10080H3 + 1512H5 + 72H7 + H9 )/512 x8 = (1680H0 + 3360H2 + 840H4 + 56H6 + H8 )/256

A-100

n! k ≈ 1.086435

Tables of Orthogonal Polynomials H0 = 1 x10 = (30240H0 + 75600H2 + 25200H4 + 2520H6 + 90H8 + H10 )/1024 H1 = 2x x9 = (15120H1 + 10080H3 + 1512H5 + 72H7 + H9 )/512 2 H2 = 4x − 2 x8 = (1680H0 + 3360H2 + 840H4 + 56H6 + H8 )/256 3 H3 = 8x − 12x x7 = (840H1 + 420H3 + 42H5 + H7 )/128 H4 = 16x4 − 48x2 + 12 x6 = (120H0 + 180H2 + 30H4 + H6 )/64 H5 = 32x5 − 160x3 + 120x x5 = (60H1 + 20H3 + H5 )/32 6 4 2 H6 = 64x − 480x + 720x − 120 x4 = (12H0 + 12H2 + H4 )/16 7 5 3 H7 = 128x − 1344x + 3360x − 1680x x3 = (6H1 + H3 )/8 H8 = 256x8 − 3584x6 + 13440x4 − 13440x2 + 1680 x2 = (2H0 + H2 )/4 H9 = 512x9 − 9216x7 + 48384x5 − 80640x3 + 30240x x = (H1 )/2 H10 = 1024x10 − 23040x8 + 161280x6 − 403200x4 + 302400x2 − 30240 1 = H0 L0 L1 L2 L3 L4 L5 L6

=1 x6 = 720L0 − 4320L1 + 10800L2 − 14400L3 + 10800L4 − 4320L5 + 720L6 = −x + 1 x5 = 120L0 − 600L1 + 1200L2 − 1200L3 + 600L4 − 120L5 2 = (x − 4x + 2)/2 x4 = 24L0 − 96L1 + 144L2 − 96L3 + 24L4 3 2 = (−x + 9x − 18x + 6)/6 x3 = 6L0 − 18L1 + 18L2 − 6L3 4 3 2 = (x − 16x + 72x − 96x + 24)/24 x2 = 2L0 − 4L1 + 2L2 = (−x5 + 25x4 − 200x3 + 600x2 − 600x + 120)/120 x = L0 − L1 = (x6 − 36x5 + 450x4 − 2400x3 + 5400x2 − 4320x + 720)/720 1 = L0

P0 = 1 x10 = (4199P0 + 16150P2 + 15504P4 + 7904P6 + 2176P8 + 256P10 )/46189 P1 = x x9 = (3315P1 + 4760P3 + 2992P5 + 960P7 + 128P9 )/12155 P2 = (3x2 − 1)/2 x8 = (715P0 + 2600P2 + 2160P4 + 832P6 + 128P8 )/6435 3 P3 = (5x − 3x)/2 x7 = (143P1 + 182P3 + 88P5 + 16P7 )/429 4 2 P4 = (35x − 30x + 3)/8 x6 = (33P0 + 110P2 + 72P4 + 16P6 )/231 5 3 P5 = (63x − 70x + 15x)/8 x5 = (27P1 + 28P3 + 8P5 )/63 P6 = (231x6 − 315x4 + 105x2 − 5)/16 x4 = (7P0 + 20P2 + 8P4 )/35 7 5 3 P7 = (429x − 693x + 315x − 35x)/16 x3 = (3P1 + 2P3 )/5 8 6 4 2 P8 = (6435x − 12012x + 6930x − 1260x + 35)/128 x2 = (P0 + 2P2 )/3 P9 = (12155x9 − 25740x7 + 18018x5 − 4620x3 + 315x)/128 x = P1 P10 = (46189x10 − 109395x8 + 90090x6 − 30030x4 + 3465x2 − 63)/256 1 = P0 T0 = 1 x10 = (126T0 + 210T2 + 120T4 + 45T6 + 10T8 + T10 )/512 T1 = x x9 = (126T1 + 84T3 + 36T5 + 9T7 + T9 )/256 T2 = 2x2 − 1 x8 = (35T0 + 56T2 + 28T4 + 8T6 + T8 )/128 3 T3 = 4x − 3x x7 = (35T1 + 21T3 + 7T5 + T7 )/64 4 2 T4 = 8x − 8x + 1 x6 = (10T0 + 15T2 + 6T4 + T6 )/32 5 3 T5 = 16x − 20x + 5x x5 = (10T1 + 5T3 + T5 )/16 T6 = 32x6 − 48x4 + 18x2 − 1 x4 = (3T0 + 4T2 + T4 )/8 7 5 3 T7 = 64x − 112x + 56x − 7x x3 = (3T1 + T3 )/4 8 6 4 2 T8 = 128x − 256x + 160x − 32x + 1 x2 = (T0 + T2 )/2 9 7 5 3 T9 = 256x − 576x + 432x − 120x + 9x x = T1 T10 = 512x10 − 1280x8 + 1120x6 − 400x4 + 50x2 − 1 1 = T0 U0 = 1 x10 = (42U0 + 90U2 + 75U4 + 35U6 + 9U8 + U10 )/1024 U1 = 2x x9 = (42U1 + 48U3 + 27U5 + 8U7 + U9 )/512 2 U2 = 4x − 1 x8 = (14U0 + 28U2 + 20U4 + 7U6 + U8 )/256 U3 = 8x3 − 4x x7 = (14U1 + 14U3 + 6U5 + U7 )/128 4 2 U4 = 16x − 12x + 1 x6 = (5U0 + 9U2 + 5U4 + U6 )/64 5 3 U5 = 32x − 32x + 6x x5 = (5U1 + 4U3 + U5 )/32 U6 = 64x6 − 80x4 + 24x2 − 1 x4 = (2U0 + 3U2 + U4 )/16 U7 = 128x7 − 192x5 + 80x3 − 8x x3 = (2U1 + U3 )/8 8 6 4 2 U8 = 256x − 448x + 240x − 40x + 1 x2 = (U0 + U2 )/4 9 7 5 3 U9 = 512x − 1024x + 672x − 160x + 10x x = (U1 )/2 U10 = 1024x10 − 2304x8 + 1792x6 − 560x4 + 60x2 − 1 1 = U0

TeamLRN

Clebsch–Gordan coefficients 

j1 m1

×

j2 j m2 m

 k

'

 = δm,m1 +m2

(j1 + j2 − j)!(j + j1 − j2 )!(j + j2 − j1 )!(2j + 1) (j + j1 + j2 + 1)!


(−1)k (j1 + m1 )!(j1 − m1 )!(j2 + m2 )!(j2 − m2 )!(j + m)!(j − m)! . k!(j1 + j2 − j − k)!(j1 − m1 − k)!(j2 + m2 − k)!(j − j2 + m1 + k)!(j − j1 − m2 + k)!

1. Conditions: (a) Each of {j1 , j2 , j, m1 , m2 , m} may be an integer, or half an integer. Additionally: j > 0, j1 > 0, j2 > 0 and j + j1 + j2 is an integer. (b) j1 + j2 − j ≥ 0. (c) j1 − j2 + j ≥ 0. (d) −j1 + j2 + j ≥ 0. (e) |m1 | ≤ j1 , |m2 | ≤ j2 , |m| ≤ j. 2. Special values:   j2 j j1 = 0 if m1 + m2 = m. (a) m m2 m   1 j1 0 j = δj1 ,j δm1 ,m . (b) m1 0 m   j 1 j2 j = 0 when j1 + j2 + j is an odd integer. (c) 0 0 0   j1 j j1 = 0 when 2j1 + j is an odd integer. (d) m1 m1 m   j1 j2 j : 3. Symmetry relations: all of the following are equal to m1 m2 m   j2 j1 j (a) , −m2 −m1 −m   j2 j1 j , (b) (−1)j1 +j2 −j m m2 m   1 j1 j2 j , (c) (−1)j1 +j2 −j −m1 −m2 −m    j j2 j1 2j+1 j2 +m2 (d) (−1) , 2j1 +1 −m m2 −m1    j j2 j1 2j+1 (−1)j1 −m1 +j−m , (e) 2j1 +1 m −m2 m1    j2 j j1 2j+1 , (−1)j−m+j1 −m1 (f) 2j1 +1 m2 −m −m1    j1 j j2 2j+1 (−1)j1 −m1 , (g) 2j2 +1 m1 −m −m2    j j1 j2 2j+1 (h) (−1)j1 −m1 . 2j2 +1 m −m1 m2 By use of the symmetry relations, Clebsch–Gordan coefficients may be put in the standard form j1 ≤ j2 ≤ j and m ≥ 0.

A-102

 m2

m

j1

j

− 12

0

1 2

1

0

1 2

1 2

1

1 2

0

1 2

1

1 2

1 2

1 2

1

1

1 2

1 2

j1 m1

√

1

m2

m

j1

j

−1

0

1

1

−1

0

1

2

− 12

0

1 2

3 2

− 12

1 2

1

1

− 12

1 2

1

2

0

0

1

2

0

0

1 2

3 2

0

1 2

1 2

3 2

0

1 2

1

1

0

1 2

1

2

0

1

1

1

2 2 √ 3 2 √ 2 2 √ 3 2

1 2 m2

j m



 m2

m

j1

j

≈ 0.707107

0

1

1

2

≈ 0.866025

1 2

0

1 2

3 2

≈ 0.707107

1 2

1 2

1

1

≈ 0.866025

1 2

1 2

1

2

1 2

1

1 2

3 2

1 2

3 2

1

2

1

0

1

1

√

1 ≈ 1.000000   j1 1 j m1 m2 m √ 2 2 √ 6 6 √ 2 2 3 4 √ 5 4 √ 6 3 √ 3 2 √ 6 3 √ 2 4 √ 10 4 √ 2 2

j1 m1

≈ 0.707107 ≈ 0.408248 ≈ 0.707107 ≈ 0.750000 ≈ 0.559017 ≈ 0.816496 ≈ 0.866025 ≈ 0.8164967 ≈ 0.353553 ≈ 0.790569 ≈ 0.707107

1

0

1

2

1

1 2

1 2

3 2

2 2 √ 2 2 √ − 42 √ 10 4 √ 30 6 √ 105 12 √ − 22 √ 6 6 √ 3 3

1

1 2

1

1

− 34

1

1 2

1

2

1

1

1 2

3 2

1

1

1

1

√

≈ 0.707107 ≈ 0.707107 ≈ −0.353553 ≈ 0.790569 ≈ 0.912871 ≈ 0.853913 ≈ −0.707107 ≈ 0.408248 ≈ 0.577350 ≈ −0.750000 ≈ 0.559017

10 4 √ − 22 √ 2 2

≈ 0.790569

1

≈ 1.000000

√

1

1

2

1

3 2

1 2

3 2

1

3 2

1

2

√ 105 12

1

2

1

2

1

TeamLRN



5 4

1

A-103

1 j m2 m

≈ −0.707107 ≈ 0.707107

≈ 0.853913 ≈ 1.000000

NORMAL PROBABILITY FUNCTION Table of the normal distribution For a standard normal random variable (Φ(z) is the area under the Standard Normal Curve from −∞ to z). Proportion of the total area (%) 68.27 90 95 95.45 99.0 99.73 99.8 99.9

Limits µ − λσ µ−σ µ − 1.65σ µ − 1.96σ µ − 2σ µ − 2.58σ µ − 3σ µ − 3.09σ µ − 3.29σ x Φ(x) 2[1 − Φ(x)] x 1 − Φ(x)

3.09 10−3

µ + λσ µ+σ µ + 1.65σ µ + 1.96σ µ + 2σ µ + 2.58σ µ + 3σ µ + 3.09σ µ + 3.29σ 1.282 0.90 0.20 3.72 10−4

1.645 0.95 0.10 4.26 10−5

1.960 0.975 0.05 4.75 10−6

Remaining area (%) 31.73 10 5 4.55 0.99 0.27 0.2 0.1

2.326 0.99 0.02 5.20 10−7

2.576 0.995 0.01 5.61 10−8

3.090 0.999 0.002 6.00 10−9

6.36 10−10

Areas under the Standard Normal Curve from 0 to z z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

0 .0000 .0398 .0793 .1179 .1554 .1915 .2258 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 4990 4993 4995 4997 4998 4998 4999 4999 5000

1 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2612 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .4991 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .5000

2 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2652 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .4991 .4994 .4995 .4997 .4998 .4999 .4999 .4999 .5000

3 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .4991 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

4 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2996 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

A-104

5 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

6 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

7 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .4992 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

8 .0319 .0714 .1103 .1480 .1844 .2190 .2518 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .4993 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

9 .0359 .0754 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .4999 .5000

Common sample size calculations Parameter

Estimate

Sample size

µ

x ¯

n=

p

pˆ

n=

µ2 − µ2

x ¯1 − x ¯2

n1 = n2 =

(zα/2 )2 (σ12 + σ22 ) E2

p1 − p2

pˆ1 − pˆ2

n1 = n2 =

(zα/2 )2 (p1 q1 + p2 q2 ) E2

z

α/2

· σ 2

E (zα/2 )2 · pq E2

Common one sample confidence intervals Parameter

Assumptions

100(1 − α)% Confidence interval

µ

n large, σ 2 known, or normality, σ 2 known

σ x ¯ ± zα/2 · √ n

µ

normality, σ 2 unknown

σ2

normality

p

binomial experiment, n large

s x ¯ ± tα/2,n−1 · √ n   (n − 1)s2 (n − 1)s2 , χ2α/2,n−1 χ21−α/2,n−1

pˆ(1 − pˆ) pˆ ± zα/2 · n

Common two sample confidence intervals 100(1 − α)% Confidence interval

Parameter

Assumptions

µ1 − µ2

normality, independence, σ12 , σ22 known or n1 , n2 large, independence, σ12 , σ22 known

(¯ x1 − x ¯2 ) ± zα/2 · (¯ x1 − x ¯2 ) ±

σ12

=

σ22

µ1 − µ2

normality, independence, unknown

µ1 − µ2

normality, independence, σ12 = σ22 unknown

σ2 σ12 + 2 n1 n2

1 1 + n1 n2 (n1 − 1)s21 + (n2 − 1)s22 s2p = n1 + n2 −2 s2 s21 (¯ x1 − x ¯2 ) ± tα/2,ν · + 2 n n 1 2  

t α2 ,n1 +n2 −2 · sp

ν≈

s2 1 n1

2 (s2 1 /n1 ) n1 −1

+ +

s2 2 n2

2

2 (s2 2 /n2 ) n2 −1

µ1 − µ2

normality, n pairs, dependence

sd d¯ ± tα/2,n−1 · √ n

p1 − p2

binomial experiments, n1 , n2 large, independence

(ˆ p1 − pˆ2 )±

pˆ2 (1 − pˆ2 ) pˆ1 (1 − pˆ1 ) zα/2 · + n1 n2

A-105

TeamLRN

PERCENTAGE POINTS, STUDENT’S t-DISTRIBUTION This table gives values of t such that F (t) =

    Γ n+1 x2 n+1 2  1 + dx − √ n n 2 nπΓ 2 −∞ for n, the number of degrees of freedom, equal to 1, 2, . . . , 30, 40, 60, 120, ∞; and for F (t) = 0.60, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, and 0.9995. The t-distribution is symmetrical, so that F (−t) = 1 − F (t) n/F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

.60 .325 .289 .277 .271 .267 .265 .263 .262 .261 .260 .260 .259 .259 .258 .258 .258 .257 .257 .257 .257 .257 .256 .256 .256 .256 .256 .256 .256 .256 .256 .255 .254 .254 .253

.75 1.000 .816 .765 .741 .727 .718 .711 .706 .703 .700 .697 .695 .694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .685 .684 .684 .684 .683 .683 .683 .681 .679 .677 .674

t

.90 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.296 1.289 1.282

.95 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 1.645

.975 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.980 1.960

.99 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.358 2.326

.995 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.617 2.576

.9995 636.619 31.598 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.767 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.373 3.291

*This table is abridged from the “Statistical Tables” of R. A. Fisher and Frank Yates published by Oliver & Boyd. Ltd., Edinburgh and London, 1938. It is here published with the kind permission of the authors and their publishers.

χ2 F (χ)2 =

χ2

1

0

x!

A-106

n/F ∞

PERCENTAGE POINTS, CHI-SQUARE DISTRIBUTION This table gives values of χ2 such that

χ2

1   x(n−2)/2 e−x/2 dx 2n/2 Γ n2 for n, the number of degrees √ of freedom, √ equal to 1, 2, . . ., 30. For n > 30, a normal approximation is quite accurate. The expression 2x2 − 2n − 1 is approximately normally distributed as the standard normal distribution. Thus χ2α , the α-point of the distribution, may be computed by the formula √ 1 χ2α = [xα + 2n − 1]2 , 2 F (χ)2 =

0

where xα is the α-point of the cumulative normal distribution. For even values of n, F (χ2 ) can be written as 1 − F (χ2 ) =

 x −1

x=0 1 2 χ 2



e−λ λx x!

1 n. 2

with λ = and x = Thus the cumulative Chi-Square distribution is related to the cumulative Poisson distribution. Another approximate formula for large n 

3 2 2 2 χα = n 1 − + zα 9n 9n n = degrees of freedom zα = the normal deviate (the value of x for which F (x) = the desired percentile). x 1.282 1.645 1.960 2.326 2.576 3.090 F (x) .90 .95 .975 .99 .995 .999 χ2.99 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 88.4 is the 99th percentile for 60 degrees of freedom.

χ2

1   xn−2/2 e−x/2 dx 2n/2 Γ n2

F (χ2 ) = 0

( n F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

.005 .0000393 .0100 .0717 .207 .412 .676 .989 1.34 1.73 2.16 2.60 3.07 3.57 4.07 4.60 5.14 5.70 6.26 6.84 7.43 8.03 8.64 9.26 9.89 10.5 11.2 11.8 12.5 13.1 13.8

.010 .000157 .0201 .115 .297 .554 .872 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54 10.2 10.9 11.5 12.2 12.9 12.6 14.3 15.0

.025 .000982 .0506 .216 .484 .831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.26 6.91 7.56 8.23 8.91 9.59 10.3 11.0 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8

.050 .00393 .103 .352 .711 1.15 1.64 2.17 2.73 3.33 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.1 10.9 11.6 12.3 13.1 13.8 14.6 15.4 16.2 16.9 17.7 18.5

.100 .0158 .211 .584 1.06 1.61 2.20 2.83 3.49 4.17 4.87 5.58 6.30 7.04 7.79 8.55 9.31 10.1 10.9 11.7 12.4 13.2 14.0 14.8 15.7 16.5 17.3 18.1 18.9 19.8 20.6

.250 .102 .575 1.21 1.92 2.67 3.45 4.25 5.07 5.90 6.74 7.58 8.44 9.30 10.2 11.0 11.9 12.8 13.7 14.6 15.5 16.3 17.2 18.1 19.0 19.9 20.8 21.7 22.7 23.6 24.5

A-106 TeamLRN

.500 .455 1.39 2.37 3.36 4.35 5.35 6.35 7.34 8.34 9.34 10.3 11.3 12.3 13.3 14.3 15.3 16.3 17.3 18.3 19.3 20.3 21.3 22.3 23.3 24.3 25.3 26.3 27.3 28.3 29.3

.750 1.32 2.77 4.11 5.39 6.63 7.84 9.04 10.2 11.4 12.5 13.7 14.8 16.0 17.1 18.2 19.4 20.5 21.6 22.7 23.8 24.9 26.0 27.1 28.2 29.3 30.4 31.5 32.6 33.7 34.8

.900 2.71 4.61 6.25 7.78 9.24 10.6 12.0 13.4 14.7 16.0 17.3 18.5 19.8 21.1 22.3 23.5 24.8 26.0 27.2 28.4 29.6 30.8 32.0 33.2 34.4 35.6 36.7 37.9 39.1 40.3

.950 3.84 5.99 7.81 9.49 11.1 12.6 14.1 15.5 16.9 18.3 19.7 21.0 22.4 23.7 25.0 26.3 27.6 28.9 30.1 31.4 32.7 33.9 35.2 36.4 37.7 38.9 40.1 41.3 42.6 43.8

.975 5.02 7.38 9.35 11.1 12.8 14.4 16.0 17.5 19.0 20.5 21.9 23.3 24.7 26.1 27.5 28.8 30.2 31.5 32.9 34.2 35.5 36.8 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0

.990 6.63 9.21 11.3 13.3 15.1 16.8 18.5 20.1 21.7 23.2 24.7 26.2 27.7 29.1 30.6 32.0 33.4 34.8 36.2 37.6 38.9 40.3 41.6 43.0 44.3 45.6 47.0 48.3 49.6 50.9

.995 7.88 10.6 12.8 14.9 16.7 18.5 20.3 22.0 23.6 25.2 26.8 28.3 29.8 31.3 32.8 34.3 35.7 37.2 38.6 40.0 41.4 42.8 44.2 45.6 46.9 48.3 49.6 51.0 52.3 53.7

PERCENTAGE POINTS, F -DISTRIBUTION This table gives values of F such that   F Γ m+n  m  2  n  mm/2 nn/2 xm−2/2 (n + mx)−(m+n)/2 dx F (F ) = Γ 2 Γ 2 0 for selected values of m, the number of degrees of freedom of the numerator of F ; and for selected values of n, the number of degrees freedom of the denominator of F . The table also provides values corresponding to F (F )=.10,.05,.025,.01,.005,.001 since F1−α for m and n degrees of freedom is the reciprocal of Fα for n and m degrees of freedom. Thus 1 1 = = .164 F.05 (4, 7) = F.95 (7, 4) 6.09

A-107



F

F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

  Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .90 Γ m Γ 2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

39.86 8.53 5.54 4.54 4.06 3.78 3.59 3.46 3.36 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.90 2.89 2.89 2.88 2.84 2.79 2.75 2.71

49.50 9.00 5.46 4.32 3.78 3.46 3.26 3.11 3.01 2.92 2.86 2.81 2.76 2.73 2.70 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.44 2.39 2.35 2.30

53.59 9.16 5.39 4.19 3.62 3.29 3.07 2.92 2.81 2.73 2.66 2.61 2.56 2.52 2.49 2.46 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.28 2.23 2.18 2.13 2.08

55.83 9.24 5.34 4.11 3.52 3.18 2.96 2.81 2.69 2.61 2.54 2.48 2.43 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.17 2.16 2.15 2.14 2.09 2.04 1.99 1.94

57.24 9.29 5.31 4.05 3.45 3.11 2.88 2.73 2.61 2.52 2.45 2.39 2.35 2.31 2.27 2.24 2.22 2.20 2.18 2.16 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.06 2.06 2.05 2.00 1.95 1.90 1.85

58.20 9.33 5.28 4.01 3.40 3.05 2.83 2.67 2.55 2.46 2.39 2.33 2.28 2.24 2.21 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02 2.01 2.00 2.00 1.99 1.98 1.93 1.87 1.82 1.77

58.91 9.35 5.27 3.98 3.37 3.01 2.78 2.62 2.51 2.41 2.34 2.28 2.23 2.19 2.16 2.13 2.10 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.87 1.82 1.77 1.72

59.44 9.37 5.25 3.95 3.34 2.98 2.75 2.59 2.47 2.38 2.30 2.24 2.20 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.83 1.77 1.72 1.67

59.86 9.38 5.24 3.94 3.32 2.96 2.72 2.56 2.44 2.35 2.27 2.21 2.16 2.12 2.09 2.06 2.03 2.00 1.98 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.87 1.86 1.85 1.79 1.74 1.68 1.63

60.19 9.39 5.23 3.92 3.30 2.94 2.70 2.54 2.42 2.32 2.25 2.19 2.14 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.88 1.87 1.86 1.85 1.84 1.83 1.82 1.76 1.71 1.65 1.60

60.71 9.41 5.22 3.90 3.27 2.90 2.67 2.50 2.38 2.28 2.21 2.15 2.10 2.05 2.02 1.99 1.96 1.93 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 1.71 1.66 1.60 1.55

61.22 9.42 5.20 3.87 3.24 2.87 2.63 2.46 2.34 2.24 2.17 2.10 2.05 2.01 1.97 1.94 1.91 1.89 1.86 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.66 1.60 1.55 1.49

61.74 9.44 5.18 3.84 3.21 2.84 2.59 2.42 2.30 2.20 2.12 2.06 2.01 1.96 1.92 1.89 1.86 1.84 1.81 1.79 1.78 1.76 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.61 1.54 1.48 1.42

62.00 9.45 5.18 3.83 3.19 2.82 2.58 2.40 2.28 2.18 2.10 2.04 1.98 1.94 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.72 1.70 1.69 1.68 1.67 1.66 1.65 1.64 1.57 1.51 1.45 1.38

62.26 9.46 5.17 3.82 3.17 2.80 2.56 2.38 2.25 2.16 2.08 2.01 1.96 1.91 1.87 1.84 1.81 1.78 1.76 1.74 1.72 1.70 1.69 1.67 1.66 1.65 1.64 1.63 1.62 1.61 1.54 1.48 1.41 1.34

62.53 9.47 5.16 3.80 3.16 2.78 2.54 2.36 2.23 2.13 2.05 1.99 1.93 1.89 1.85 1.81 1.78 1.75 1.73 1.71 1.69 1.67 1.66 1.64 1.63 1.61 1.60 1.59 1.58 1.57 1.51 1.44 1.37 1.30

62.79 9.47 5.15 3.79 3.14 2.76 2.51 2.34 2.21 2.11 2.03 1.96 1.90 1.86 1.82 1.78 1.75 1.72 1.70 1.68 1.66 1.64 1.62 1.61 1.59 1.58 1.57 1.56 1.55 1.54 1.47 1.40 1.32 1.24

63.06 9.48 5.14 3.78 3.12 2.74 2.49 2.32 2.18 2.08 2.00 1.93 1.88 1.83 1.79 1.75 1.72 1.69 1.67 1.64 1.62 1.60 1.59 1.57 1.56 1.54 1.53 1.52 1.51 1.50 1.42 1.35 1.26 1.17

63.33 9.49 5.13 3.76 3.10 2.72 2.47 2.29 2.16 2.06 1.97 1.90 1.85 1.80 1.76 1.72 1.69 1.66 1.63 1.61 1.59 1.57 1.55 1.53 1.52 1.50 1.49 1.48 1.47 1.46 1.38 1.29 1.19 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on mandndegrees of freedom, respectively.   F Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .95 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

161.4 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.00 3.92 3.84

199.5 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.15 3.07 3.00

215.7 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.76 2.68 2.60

224.6 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.53 2.45 2.37

230.2 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.37 2.29 2.21

234.0 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.25 2.17 2.10

236.8 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.17 2.09 2.01

238.9 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.10 2.02. 1.94

240.5 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.04 1.96 1.88

241.9 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 1.99 1.91 1.83

243.9 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 1.92 1.83 1.75

245.9 19.43 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.84 1.75 1.67

248.0 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.84 1.75 1.66 1.57

249.1 19.45 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.95 1.93 1.91 1.90 1.89 1.79 1.70 1.61 1.52

250.1 19.46 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.90 1.88 1.87 1.85 1.84 1.74 1.65 1.55 1.46

251.1 19.47 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.85 1.84 1.82 1.81 1.79 1.69 1.59 1.50 1.39

252.2 19.48 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.79 1.77 1.75 1.74 1.64 1.53 1.43 1.32

253.3 19.49 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.71 1.70 1.68 1.58 1.47 1.35 1.22

254.3 19.50 8.53 5.63 4.36 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 139 1.25 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on m and n degrees of freedom, respectively.

A-108

TeamLRN



F

F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

  Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .975 Γ m Γ 2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

647.8 38.51 17.44 12.22 10.01 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.41 6.30 6.20 6.12 6.04 5.98 5.92 5.87 5.83 5.79 5.75 5.72 5.69 5.66 5.63 5.61 5.59 5.57 5.42 5.29 5.15 5.02

799.5 39.00 16.04 10.65 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10 4.97 4.86 4.77 4.69 4.62 4.56 4.51 4.46 4.42 4.38 4.35 4.32 4.29 4.27 4.24 4.22 4.20 4.18 4.05 3.93 3.80 3.69

864.2 39.17 15.44 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.01 3.95 3.90 3.86 3.82 3.78 3.75 3.72 3.69 3.67 3.65 3.63 3.61 3.59 3.46 3.34 3.23 3.12

899.6 39.25 15.10 9.60 7.39 6.23 5.52 5.05 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56 3.51 3.48 3.44 3.41 3.38 3.35 3.33 3.31 3.29 3.27 3.25 3.13 3.01 2.89 2.79

921.8 39.30 14.88 9.36 7.15 5.99 5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58 3.50 3.44 3.38 3.33 3.29 3.25 3.22 3.18 3.15 3.13 3.10 3.08 3.06 3.04 3.03 2.90 2.79 2.67 2.57

937.1 39.33 14.73 9.20 6.98 5.82 5.12 4.65 4.32 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17 3.13 3.09 3.05 3.02 2.99 2.97 2.94 2.92 2.90 2.88 2.87 2.74 2.63 2.52 2.41

948.2 39.36 14.62 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05 3.01 2.97 2.93 2.90 2.87 2.85 2.82 2.80 2.78 2.76 2.75 2.62 2.51 2.39 2.29

956.7 39.37 14.54 8.98 6.76 5.60 4.90 4.43 4.10 3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01 2.96 2.91 2.87 2.84 2.81 2.78 2.75 2.73 2.71 2.69 2.67 2.65 2.53 2.41 2.30 2.19

963.3 39.39 14.47 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88 2.84 2.80 2.76 2.73 2.70 2.68 2.65 2.63 2.61 2.59 2.57 2.45 2.33 2.22 2.11

968.6 39.40 14.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82 2.77 2.73 2.70 2.67 2.64 2.61 2.59 2.57 2.55 2.53 2.51 2.39 2.27 2.16 2.05

976.7 39.41 14.34 8.75 6.52 5.37 4.67 4.20 3.87 3.62 3.43 3.28 3.15 3.05 2.96 2.89 2.82 2.77 2.72 2.68 2.64 2.60 2.57 2.54 2.51 2.49 2.47 2.45 2.43 2.41 2.29 2.17 2.05 1.94

984.9 39.43 14.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3.33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62 2.57 2.53 2.50 2.47 2.44 2.41 2.39 2.36 2.34 2.32 2.31 2.18 2.06 1.94 1.83

993.1 39.45 14.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.68 2.62 2.56 2.51 2.46 2.42 2.39 2.36 2.33 2.30 2.28 2.25 2.23 2.21 2.20 2.07 1.94 1.82 1.71

997.2 39.46 14.12 8.51 6.28 5.12 4.42 3.95 3.61 3.37 3.17 3.02 2.89 2.79 2.70 2.63 2.56 2.50 2.45 2.41 2.37 2.33 2.30 2.27 2.24 2.22 2.19 2.17 2.15 2.14 2.01 1.88 1.76 1.64

1001 39.46 14.08 8.46 6.23 5.07 4.36 3.89 3.56 3.31 3.12 2.96 2.84 2.73 2.64 2.57 2.50 2.44 2.39 2.35 2.31 2.27 2.24 2.21 2.18 2.16 2.13 2.11 2.09 2.07 1.94 1.82 1.69 1.57

1006 39.47 14.04 8.41 6.18 5.01 4.31 3.84 3.51 3.26 3.06 2.91 2.78 2.67 2.59 2.51 2.44 2.38 2.33 2.29 2.25 2.21 2.18 2.15 2.12 2.09 2.03 2.05 2.03 2.01 1.88 1.74 1.61 1.48

1010 39.48 13.99 8.36 6.12 4.96 4.25 3.78 3.45 3.20 3.00 2.85 2.72 2.61 2.52 2.45 2.38 2.32 2.27 2.22 2.18 2.14 2.11 2.08 2.05 2.03 2.00 1.98 1.96 1.94 1.80 1.67 1.53 1.39

1014 39.49 13.95 8.31 6.07 4.90 4.20 3.73 3.39 3.14 2.94 2.79 2.66 2.55 2.46 2.38 2.32 2.26 2.20 2.16 2.11 2.08 2.04 2.01 1.98 1.95 1.93 1.91 1.89 1.87 1.72 1.58 1.43 1.27

1018 39.50 13.90 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.13 2.09 2.04 2.00 1.97 1.94 1.91 1.88 1.85 1.83 1.81 1.79 1.64 1.48 1.31 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on mandndegrees of freedom, respectively.   F Γ m+n 2    n  mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .99 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

F =

s2 1 s2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

4052 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7.88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.08 6.85 6.63

4999.5 99.00 30.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 4.98 4.79 4.61

5403 99.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.13 3.95 3.78

5625 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.65 3.48 3.32

5764 99.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.85 3.82 3.78 3.75 3.73 3.70 3.51 3.34 3.17 3.02

5859 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.12 2.96 2.80

5928 99.36 27.67 14.98 10.46 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.42 3.39 3.36 3.33 3.30 3.12 2.95 2.79 2.64

5982 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.82 2.66 2.51

6022 99.39 27.35 14.66 10.16 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3.18 3.15 3.12 3.09 3.07 2.89 2.72 2.56 2.41

6056 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.31 3.26 3.21 3.17 3.13 3.09 3.06 3.03 3.00 2.98 2.80 2.63 2.47 2.32

6106 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.50 2.34 2.18

6157 99.43 26.87 14.20 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.81 2.78 2.75 2.73 2.70 2.52 2.35 2.19 2.04

6209 99.45 26.69 14.02 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.88 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.37 2.20 2.03 1.88

6235 99.46 26.60 13.93 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.80 2.75 2.70 2.66 2.62 2.58 2.55 2.52 2.49 2.47 2.29 2.12 1.95 1.79

6261 99.47 26.50 13.84 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.54 2.50 2.47 2.44 2.41 2.39 2.20 2.03 1.86 1.70

6287 99.47 26.41 13.75 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.30 2.11 1.94 1.76 1.59

6313 99.48 26.32 13.65 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2.55 2.50 2.45 2.40 2.36 2.33 2.29 2.26 2.23 2.21 2.02 1.84 1.66 1.47

6339 99.49 26.22 13.56 9.11 6.97 5.74 4.95 4.40 4.00 3.69 3.45 3.25 3.09 2.96 2.84 2.75 2.66 2.58 2.52 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.17 2.14 2.11 1.92 1.73 1.53 1.32

6366 99.50 26.13 13.46 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.06 2.03 2.01 1.80 1.60 1.38 1.00

=

S1 S2 /n, m

where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common

2

variance σ and based on m and n degrees of freedom, respectively.

A-109

APPENDIX B: SOURCES OF PHYSICAL AND CHEMICAL DATA

TeamLRN

SOURCES OF PHYSICAL AND CHEMICAL DATA In addition to the primary research journals, there are many useful sources of property data of the type contained in the CRC Handbook of Chemistry and Physics. A selected list of these is presented here, with emphasis on print and electronic sources whose contents have been subject to a reasonable level of quality control.

A. Data Journals 1. Journal of Physical and Chemical Reference Data – Published jointly by the National Institute of Standards and Technology and the American Institute of Physics, this quarterly journal contains compilations of evaluated data in chemistry, physics, and materials science. It is available in print and on the Internet. [ojps.aip.org/jpcrd/] 2. Journal of Chemical and Engineering Data – This bimonthly journal of the American Chemical Society publishes articles reporting original experimental measurements carried out under carefully controlled conditions. The main emphasis is on thermochemical and thermophysical properties. Review articles with evaluated data from the literature are also published. [pubs.acs.org/journals/jceaax/index.html] 3. Journal of Chemical Thermodynamics – This journal publishes original research papers that include highly accurate measurements of thermodynamic and thermophysical properties. [http://www.sciencedirect.com] 4. Atomic Data and Nuclear Data Tables – This is a bimonthly journal containing compilations of data in atomic physics, nuclear physics, and related fields. [www.sciencedirect.com] 5. Journal of Phase Equilibria and Diffusion – This journal presents critically evaluated phase diagrams and related data on alloy systems. It is published by ASM International and is the successor to the previous ASM periodical Bulletin Of Alloy Phase Diagrams. [www.asm-intl.org.] 6. Journal of Chemical Information and Computer Sciences – Although not a true data journal, it contains many papers on the prediction of physical property data from molecular structure. It is published by the American Chemical Society. [pubs.acs.org/journals/jcisd8/index.html]

B. Data Centers This section lists selected organizations that perform a continuing function of compiling and critically evaluating data in specific fields of science. 1. National Institute of Standards and Technology – Under its Standard Reference Data program, NIST supports a number of data centers in chemistry, physics, and materials science. Topics covered include thermodynamics, fluid properties, chemical kinetics, mass spectroscopy, atomic spectroscopy, fundamental physical constants, ceramics, and crystallography. Address: Office of Standard Reference Data, National Institute of Standards and Technology, Gaithersburg, MD 20899 [www.nist.gov/srd/]. 2. Thermodynamics Research Center – Now located at the National Institute of Standards and Technology, TRC maintains an extensive archive of data covering thermodynamic, thermochemical, and transport properties of organic compounds and mixtures. Data are distributed in both print and electronic form. Address: Mailcode 838.00, 325 Broadway, Boulder, CO 80305-3328 [www.trc.nist.gov] . 3. Design Institute for Physical Property Data – Under the auspices of the American Institute of Chemical Engineers [www.aiche.org/dippr/], DIPPR offers evaluated data on industrially-important chemical compounds. The largest project deals with physical, thermodynamic, and transport properties of pure compounds. Address: Brigham Young University, Provo, UT 84602 [dippr.byu.edu] . 4. Dortmund Data Bank – Maintains extensive databases on thermodynamic and transport properties of pure compounds and mixtures of industrial interest. The data are distributed through DECHEMA, FIZ CHEMIE, and other outlets. An abbreviated database system is also available for educational use. Address: DDBST GmbH, Industriestr. 1, 26121 Oldenburg, Germany [www.ddbst.de]. 5. Cambridge Crystallographic Data Centre – Maintains the Cambridge Structural Database of over 250,000 organic compounds. The data files and manipulation software are distributed in several ways. Address: 12 Union Rd., Cambridge CB2 1EZ, UK [www.ccdc.cam.ac.uk]. 6. FIZ Karlsruhe – In addition to many bibliographic databases, FIZ Karlsruhe maintains the Inorganic Crystal Structure Database in collaboration with the National Institute of Standards and Technology. The ICSD contains the atomic coordinates and related data on over 50,000 inorganic crystals. Address: Fachinformationszentrum (FIZ) Karlsruhe, Hermann-von-Helmholtz-Platz 1, D-76344 EggensteinLeopoldshafen, Germany [crystal.fiz-karlsruhe.de]. 7. International Centre for Diffraction Data – Maintains and distributes the Powder Diffraction File (PDF), a file of x-ray powder diffraction patterns used for identification of crystalline materials. The ICDD also distributes the NIST Crystal Data file, which contains lattice parameters for over 235,000 inorganic and organic crystalline materials. Address: 12 Campus Blvd., Newton Square, PA 19073-3273 [icdd.com]. 8. Research Collaboratory for Structural Bioinformatics – Maintains the Protein Data Bank (PDB), a file of 3-dimensional structures of proteins and other biological macromolecules. Address: Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, NJ 08854-8087 [www.rcsb.org]. 9. Toth Information Systems – Maintains the Metals Crystallographic Data File (CRYSTMET). Address: 2045 Quincy Ave., Gloucester, ON, Canada K1J 6B2 [www.tothcanada.com]. 10. Atomic Mass Data Center – Collects and evaluates high-precision data on masses of individual isotopes and maintains a comprehensive database. Address: C.S.N.S.M (IN2P3-CNRS), Batiment 108, F-91405 Orsay Campus, France [csnwww.in2p3.fr/amdc/]. 11. Particle Data Group – International center for data of high-energy physics; maintains database of properties of fundamental particles, which is published in both print and electronic form. Address: MS 50-308, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 [pdg.lbl.gov]. 12. National Nuclear Data Center – Maintains databases on nuclear structure and reactions, including neutron cross sections. The NNDC is the U. S. node in an international network of nuclear data centers. Address: Brookhaven National Laboratory, Upton, NY 11973-5000 [www.nndc.bnl.gov].

B-1

SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) 13. International Union of Pure and Applied Chemistry – Address: PO Box 13757, Research Triangle Park, NC 27709-3757 [www.iupac.org]. IUPAC supports a number of long-term data projects, including these examples: a. Solubility Data Project – Carries out evaluation of all types of solubility data. The results are published in the Solubility Data Series, whose current outlet is the Journal of Physical and Chemical Reference Data. [www.unileoben.ac.at/~eschedor/] b. Kinetic Data for Atmospheric Chemistry – Maintains a comprehensive database on the kinetics of reactions important in the chemistry of the atmosphere. [www.iupac-kinetic.ch.cam.ac.uk/] c. International Thermodynamic Tables for the Fluid State – Prepares definitive tables of the thermodynamic properties of industrially important fluids. Thirteen volumes have been published by IUPAC. [http://www.iupac.org/publications/books/seriestitles/]

C. Major Multi-Volume Handbook Series 1. Chapman & Hall/CRC Chemical Dictionaries – These originally appeared in print form as the Dictionary of Organic Compounds, Dictionary of Natural Products, etc. They are now published in electronic form and are available in CDROM format [www.crcpress.com] and on the Internet [www.chemnetbase.com]. The consolidated version, called the Combined Chemical Dictionary, has data on more than 450,000 compounds spanning all branches of chemistry. The coverage includes physical properties, biological sources, hazard information, uses, and literature references. 2. Properties of Organic Compounds – Originally published in three editions as the Handbook of Data on Organic Compounds, it is now in electronic form as Properties of Organic Compounds. The database includes about 30,000 compounds; physical properties and spectral data (mass, infrared, Raman, ultraviolet, and NMR) are covered. It is offered as CDROM [www.crcpress.com] and web access [www.chemnetbase.com]. 3. Beilstein Handbook of Organic Chemistry – The classic source of data on organic compounds, dating from the 18th century, Beilstein was converted to electronic form in the last decade of the 20th century. Over 8 million compounds and 5 million chemical reactions are now covered, with a broad range of physical properties as well as synthetic methods and ecological data. The database is accessed by the CrossFire software [www.mdli.com]. 4. Gmelin Handbook of Inorganic and Organometallic Chemistry – A subset of the information in the print series has been converted to electronic form and is now distributed in the same manner as Beilstein. In addition to the standard physical properties, the coverage includes a wide range of optical, magnetic, spectroscopic, thermal, and transport properties for about 1.4 million compounds [www.mdli.com]. 5. DECHEMA Chemical Data Series – DECHEMA distributes the DETHERM database, which emphasizes data used in process design in the chemical industry, including thermodynamic and transport properties of about 20,000 pure compounds and 90,000 mixtures. Access is available through in-house databases and via the Internet. [www.dechema.de]. 6. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology - Landolt-Börnstein covers a very broad range of data in physics, chemistry, crystallography, materials science, biophysics, astronomy, and geophysics. Hard-copy volumes in the New Series (started in 1961) are still being published, and the entire New Series is now accessible on the Internet [www.landolt-boernstein.com].

D. Selected Single-Volume Handbooks The following handbooks offer broad coverage of high-quality data in a single volume. This list is only representative; an extensive listing of handbooks in all fields of science may be found in Handbooks and Tables in Science and Technology, Third Edition (Russell H. Powell, ed., Oryx Press, Westport, CT, 1994). 1. American Institute of Physics Handbook – Although an old book, it contains much data that is still useful, especially in acoustics, mechanics, optics, and solid state physics. (Dwight E. Gray, ed., McGraw-Hill, New York, 1972) 2. Constants of Inorganic Substances - This book presents physical constants, thermodynamic data, solubility, reactivity, and other information on over 3000 inorganic compounds. Since it draws heavily on Russian literature, it contains a great deal of data that does not make its way into most U. S. handbooks. (R. A. Lidin, L. L. Andreeva, and V. A. Molochko, Begell House, New York, 1995) 3. Handbook of Chemistry and Physics – Now in the 84th Edition, the CRC Handbook covers data from most branches of chemistry and physics. The annual revisions permit regular updating of the information. Also available on CDROM [www.crcpress.com] and the web [hbcpnetbase.com]. (David R. Lide, ed., CRC Press, Boca Raton, FL, 2002) 4. Handbook of Inorganic Compounds – This book covers physical constants and solubility for about 3300 inorganic compounds. Also available on CDROM [www.crcpress.com]. (Dale L. Perry and Sidney L. Phillips, eds., CRC Press, Boca Raton, FL, 1995) 5. Handbook of Physical Properties of Liquids and Gases – This is a valuable source of data on all types of fluids, ranging from liquid and gaseous hydrocarbons to molten metals and ionized gases. Detailed tables of physical, thermodynamic, and transport properties are given for temperatures from the cryogenic region to 6000 K. Both Western and Russian literature is covered. (N. B. Vargaftik, Y. K. Vinogradov, and V. S. Yargin, Begell House, New York, 1996) 6. Handbook of Physical Quantities – The range of coverage is somewhat similar to the CRC Handbook of Chemistry and Physics, but with a stronger emphasis on physics than on chemistry. Solid state physics, lasers, nuclear physics, geophysics, and astronomy receive considerable attention. (Igor S. Grigoriev and Evgenii Z. Meilikhov, eds., CRC Press, Boca Raton, FL, 1997) 7. Kaye & Laby Tables of Physical and Chemical Constants – Kaye & Laby dates from 1911, and the 16th Edition was prepared in 1995 by a committee of experts. The coverage extends to almost every field of physics and chemistry; data on a limited number of representative substances or materials are given for each topic. (Longman Group Limited, Harlow, Essex, UK, 1995)

B-2

TeamLRN

SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) 8. Lange’s Handbook of Chemistry – Provides broad coverage of chemical data; last updated in 1998. Also available on the web [www.knovel.com]. (John A. Dean, ed., McGraw-Hill, New York, 1998) 9. Recommended Reference Materials for the Realization of Physicochemical Properties – This IUPAC book emphasizes highly accurate data on substances and materials that can be used as calibration standards. It covers physical, thermal, optical, and electrical properties. (K. N. Marsh, ed., Blackwell Scientific Publications, Oxford, 1987) 9. The Merck Index – Now in its 13th Edition (published in 2001), The Merck Index is a widely used source of data on over 10,000 compounds, chosen particularly for their importance in biology, medicine, and ecology. A short monograph on each compound gives information on the synthesis and uses as well as physical and toxicological properties. Also available on CDROM [www.camsoft.com]. (Maryadele J. O’Neil, ed., Merck & Co., Whitehouse Station, NJ, 2001)

E. Summary of Useful Web Sites for Physical and Chemical Properties Most of the web sites in the following list provide direct access to factual data on physical and chemical properties. However, the list also includes portals that link to different property databases or describe the procedure for gaining access to electronic sources of property data. There are also a few chemical directory sites, which are useful for obtaining formulas, synonyms, and registry numbers for substances of interest. Web Site

Address

Comments

Acronyms and Symbols Advanced Chemistry Development

www3.interscience.wiley.com/stasa/ www.acdlabs.com

Alloy Center

www.asminternational.org/ alloycenter/ www.geo.arizona.edu/AMS/ amcsd.php csnwww.in2p3.fr/amdc/ www.mdli.com www.ccdc.cam.ac.uk www.chemnetbase.com/scripts/ ccdweb.exe www.chemfinder.com www.oscar.chem.indiana.edu/cfdocs/ libchem/acronyms/ acronymsearch.html chem.sis.nlm.nih.gov/chemidplus/ www.chemindustry.com/chemicals/ www.chemnetbase.com

American Mineralogist Crystal Structure Database Atomic Mass Data Center Beilstein Cambridge Structural Database Chapman & Hall/CRC Combined Chemical Dictionary Chemfinder Chemical Acronyms Database

ChemIDplus ChemIndustry CHEMnetBASE ChemWeb Databases Coblentz Infrared Spectra CODATA Home Page Crystallography Open Database (COD) DECHEMA (DETHERM) DIPPR Pure Compound Database Dortmund Data Bank Enzyme Nomenclature Database FDM Reference Spectra Databases FIZ Chemie Berlin FIZ Karlsruhe - ICSD Fundamental Physical Constants Gmelin Handbook of Chemistry and Physics Hazardous Substances Data Bank IUPAC Home Page IUPAC Kinetics Data IUPAC Nomenclature Rules IUPAC Solubility Data Project Knovel.com

www.chemweb.com/databases/ www.galactic.com/coblentz/ www.codata.org www.crystallography.net www.dechema.de dippr.byu.edu www.ddbst.de www.expasy.ch/enzyme/ www.fdmspectra.com/ www.fiz-chemie.de crystal.fiz-karlsruhe.de physics.nist.gov/constants www.mdli.com hbcpnetbase.com toxnet.nlm.nih.gov/cgi-bin/sis/ htmlgen?HSDB www.iupac.org www.iupac-kinetic.ch.cam.ac.uk/ www.chem.qmul.ac.uk/iupac/ www.unileoben.ac.at/~eschedor/ www.knovel.com

B-3

Free servcie; useful for indentifying acronyms for chemicals Chemical directory, with programs for estimating physical and spectral properties Physical, electrical, thermal, and mechanical properties of alloys Lattice constants of minerals See B.10 See C.3 See B.5 See C.1 Chemical directory, with links to several property databases Useful for associating chemical names and acronyms

Chemical directory Chemical directory Portal to C&H/CRC Chemical Dictionaries, Handbook of Chemistry and Physics, Properties of Organic Compounds, etc. Portal to many databases IR spectra on CDROM Thermodynamic key values and fundamental constants Crystal data on 12,000 compounds See C.5 See B.3 See B.4 IUBMB nomenclature for enzymes Infrared, Raman, and mass spectra Portal to DETHERM (C.5) and Dortmund Data Bank (B.4) See B.6 CODATA fundamental constants See C.4 Web version of CRC Handbook Physical and toxicological properties of chemicals of health or environmental importance See B.13 See B.13.b Useful site for organic and biochemical nomenclature See B.13.a Portal to Lange’s Handbook, Perry’s Chemical Engineers’ Handbook, etc.

SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) Web Site Landolt-Börnstein MatWeb

Address www.landolt-boernstein.com www.matweb.com

Metals Crystallographic Data File NASA Chemical Kinetics Data National Center for Biotechnology Information National Nuclear Data Center National Toxicology Program NIST Atomic Spectra Database

www.tothcanada.com jpldataeval.jpl.nasa.gov www.ncbi.nlm.nih.gov

NIST Ceramics Webbook NIST Chemistry Webbook NIST Data Gateway NIST Physical Reference Data NLM Gateway NMR Shift DB Particle Data Group Polymers — A Property Database Powder Diffraction File Properties of Organic Compounds Protein Data Bank SpecInfo Spectra Online STN Easy STN Easy-Europe STN Easy-Japan Syracuse Research Corporation Table of Isotopes Thermodynamics Research Center TOXNET Wiley Interscience

Comments See C.6 Thermal, electrical, and mechanical properties of engineering materials See B.9 Kinetic and photochemical data for stratospheric modeling Portal to GenBank and other sequence databases

www.nndc.bnl.gov ntp-server.niehs.nih.gov physics.nist.gov/cgi-bin/AtData/ main_asd www.ceramics.nist.gov/webbook/ webbook.htm webbook.nist.gov srdata.nist.gov/gateway/ physics.nist.gov/PhysRefData/ gateway.nlm.nih.gov/gw/Cmd www.nmrshiftdb.org pdg.lbl.gov www.polymersdatabase.com/ icdd.com www.chemnetbase.com/scripts/ pocweb.exe www.rcsb.org www.chemicalconcepts.com spectra.galactic.com/SpectraOnline/ stneasy.cas.org stneasy.fiz-karlsruhe.de stneasy-japan.cas.org esc.syrres.com/interkow/database.htm ie.lbl.gov/education/isotopes.htm www.trc.nist.gov toxnet.nlm.nih.gov www3.interscience.wiley.com/ reference.html

See B.12 Chemical health and safety data Energy levels, wavelengths, and transition probabilities of atoms and atomic ions See B.1 Broad range of physical, thermal, and spectral properties Portal to all NIST data systems; see B.1 Atomic and molecular spectra, cross sections, x-ray attenuation, and dosimetry data Portal to all National Library of Medicine databases NMR data submitted by users See B.11 Properties of commercial polymers See B.7 See C.2 See B.8 IR, NMR, and mass spectra IR, UV, NMR, Raman, and mass spectra (unreviewed) Chemical directory (and access to Chemical Abstracts)

Properties of environmental interest Nuclear energy levels, moments, and other properties See B.2 Portal to HSDB and other databases on hazardous chemicals Portal to Kirk-Othmer Encyclopedia of Chemical Technology, Ullmann’s Encyclopedia of Industrial Chemistry, Encyclopedia of Reagents for Organic Synthesis, etc.

B-4

TeamLRN