POCKETBOOKOF MATHEMATICAL FUNCTIONS

Abridged edition of

Handbook of Mathematical Functions Milton Abramowitz and Irene A. Stegun (eds.)

Material selected by Michael Danos and Johann Rafelski

1984

Verlag Harri Deutsch - Thun - Frankfurt/Main

CONTENTS Forewordto the Original NBS Handbook 5 Pref ace 6

2. PHYSICAL CONSTANTS AND CONVERSION FACTORS 17 A.G. McNish, revised by the editors

Table 2.1. Common Units and Conversion Factors 17 Table 2.2. Names and Conversion Factors

for Electric and Magnetic Units 17 Table 2.3. AdjustedValuesof Constants 18 Table 2.4. Miscellaneous Conversion Factors 19 Table 2.5. FactorsforConvert ingCustomaryU.S. Uni t s toSIUni t s 19 Table 2.6. Geodetic Constants 19 Table 2.7. Physical andNumericalConstants 20 Table 2.8. Periodic Table of the Elements 21 Table 2.9. Electromagnetic Relations 22 Table 2.10. Radioactivity and Radiation Protection 22

3. ELEMENTARY ANALYTICAL METHODS 23 Milton Abramowitz

3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometrie Progressions; Arithmetic, Geometrie, Harmonie and Generalized Means 23

3.2. Inequalities 23 3.3. Rules for Differentiation and Integration 24 3.4. Limits, Maxima and Minima 26 3.5. Absolute and Relative Errors 27 3.6. Infinite Series 27 3.7. Complex Numbers and Functions 29 3.8. Algebraic Equations 30 3.9. Successive Approximation Methods 31 3.10. TheoremsonContinuedFractions 32 4. ELEMENTARY TRANSCENDENTAL FUNCTIONS 33

Logarithmic, Exponential, Circular and Hyperbolic Functions Ruth Zucker

4.1. Logarithmic Function 33 4.2. Exponential Function 35 4.3. Circular Functions 37 4.4. Inverse Circular Functions 45 4.5. Hyperbolic Functions 49 4.6. InverseHyberbolicFunctions 52

5. EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS .... 56 Walter Gautschi and William F. Cahill

5.1. Exponential Integral 56 5.2. Sine and Cosine Integrals 59 Table 5.1. Sine, Cosine and Exponential Integrals (0<x<10) 62 Table 5.2. Sine, Cosine and Exponential Integrals for Large

Arguments (10<x<°°) 67 Table 5.3. Sine and Cosine Integrals for Arguments nx 68 Table 5.4. Exponential Integrals E„(x)(0<x<2) 69 Table 5.5. Exponential Integrals En(x)(2<x<°°) 72 Table 5.6. Exponential Integral for Complex Arguments 73 Table 5.7. Exponential Integral for Small Complex Arguments

(lzl<5) 75

6. GAMMA FUNCTION AND RELATED FUNCTIONS 76 Philip J. Davis

6.1. Gamma Function 76 6.2. Beta Function 79 6.3. Psi(Digamma) Function 79 6.4. Polygamma Functions 81 6.5. Incomplete Gamma Function 81 6.6. Incomplete Beta Function 83

7. ERROR FUNCTION AND FRESNEL INTEGRALS 84 Walter Gautschi

7.1. Error Function 84 7.2. Repeated Integrals ofthe Error Function 86 7.3. Fresnel Integrals 87 7.4. Definite and Indefinite Integrals 89 Table 7.7 Fresnel Integrals (0<x<5) 92

8. LEGENDRE FUNCTIONS 94 Irene A. Stegun

8.1. Differential Equation 94 8.2. Relations Between LegendreFunctions 95 8.3. ValuesontheCut 95 8.4. Explicit Expressions 95 8.5. Recurrence Relations 95 8.6. Special Values 96 8.7. Trigonometrie Expansions 97 8.8. Integral Representations 97 8.9. Summation Formulas 97 8.10. Asymptotic Expansions 97 8.11. Toroidal Functions 98 8.12. ConicalFunctions 99 8.13. RelationtoEllipticIntegrals 99 8.14. Integrals 99

9. BESSEL FUNCTIONS OF INTEGER ORDER 102 F. W. J. Olver Bessel Functions J and Y 102

9.1. Def initions and Elementary Properties 102 9.2. Asymptotic Expansions for Large Arguments 108 9.3. Asymptotic Expansions for Large Orders 109 9.4. Polynomial Approximations 113 9.5. Zeros 114

Modified Bessel Functions I and K 118 9.6. Def initions and Properties 118 9.7. Asymptotic Expansions 121 9.8. Polynomial Approximations 122

Kelvin Functions 123 9.9. Definitions and Properties 123 9.10. Asymptotic Expansions 125 9.11. Polynomial Approximations 128 Table 9.1. Bessel Functions— Orders0, l ,and2(0<x<15) 130 Table 9.2. Bessel Functions — Orders 3—9(0<x<20) 136 Table 9.5. Zeros and Associated Values of Bessel Functions

and Their Derivatives (0<n<8, l<s<20) 140 Table 9.8. Modified Bessel Functions of Orders 0,1, and 2 144

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Table 9.9. ModifiedBesselFunctions —Orders 3—9(0 <x<10) 148 Table 9.12. Kelvin Functions—Orders 0 and 1(0<x<5) 150

Kelvin Functions —Modulus and Phase (0<x<7) 152

10. BESSEL FUNCTIONS OF FRACTIONAL ORDER 154 H. A. Antosiewicz

10.1. Spherical Bessel Functions 154 10.2. Modified Spherical Bessel Functions 159 10.3. Riccati-Bessel Functions 161 10.4. Airy Functions 162 Table 10.11. AiryFunctions(0<x<10) 169 Table 10.12. Integrals of Airy Functions (0<x<10) 172 Table 10.13. Zeros and Associated Values of Airy Functions

and Their Derivatives (l<s<10) 172

11. INTEGRALS OF BESSEL FUNCTIONS 173 Yudell L. Luke

11.1 Simple Integrals of Bessel Functions 173 11.2. Repeated Integrals of Jn(z) and K0(z) 175 11.3. Reduction Formulas for Indefinite Integrals 176 11.4. Definite Integrals 178 Table 11.1. Integrals of Bessel Functions 182 Table 11.2. Integrals of Bessel Functions 184

12. STRUVE FUNCTIONS AND RELATED FUNCTIONS 185 Milton Abramowitz

12.1. Struve Function H„(z) 185 12.2. Modified Struve Function L„(z) 187 12.3. Anger and Weber Functions 187 Table 12.1. Struve Functions (0<x<°o) 188

13. CONFLUENT HYPERGEOMETRIC FUNCTIONS 189 Lucy Joan Slater

13.1. Definitions of Kummer and Whittaker Functions 189 13.2. Integral Representations 190 13.3. Connections with Bessel Functions 191 13.4. Recurrence Relations and Differential Properties 191 13.5. Asymptotic Expansions and Limiting Forms 193 13.6. Special Cases 194 13.7. Zeros and Turning Values 195

14. COULOMB WAVE FUNCTIONS 198 Milton Abramowitz

14.1. Differential Equation, Series Expansions 198 14.2. Recurrence and Wronskian Relations 199 14.3. Integral Representations 199 14.4. Bessel Function Expansions 199 14.5. Asymptotic Expansions.. . . ̂ 200 14.6. Special Values and Asymptotic Behavior _ 202 14.7. Use and Extension of the Tables 203 Table 14.1. Coulomb Wave Functions of Order Zero 204 Table 14.2. C0fa) = e-">/2ir(l + ir,)l 212 15. HYPERGEOMETRIC FUNCTIONS 213

Fritz Oberhettinger 15.1. Gauss Series, Special Elementary Cases, Special Values of the

Argument 213

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15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions 214

15.3. Integral Representations and Transformation Formulas 215 15.4. Special casesof F(a, b;c;z),Polynomials and Legendre Functions 218 15.5. The Hypergeometric Differential Equation 219 15.6. Riemann's Differential Equation 221 15.7. AsymptoticExpansions i 222

16. JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS 223 L. M. Milne-Thomson

16.1 Introduction 223 16.2. Classification of the Twelve Jacobian Elliptic Functions 224 16.3. Relation of the Jacobian Functions to the Copolar Trio 224 16.4. Calculation of the Jacobian Functions by Use of the

Arithmetic-GeometricMean(A. G. M.) 225 16.5. Special Arguments 225 16.6. Jacobian Functions whenm = 0 o r l 225 16.7. Principal Terms 226 16.8. Changeof Argument 226 16.9. Relations Betweenthe Squares of the Functions 227 16.10. Change of Parameter 227 16.11. Reciprocal Parameter (Jacobi's Real Transformation) 227 16.12. Descending Landen Transformation (Gauss' Transformation) 227 16.13. Approximation in Terms of Circular Functions 227 16.14. Ascending Landen Transformation 227 16.15. Approximation in Terms of Hyperbolic Functions 228 16.16. Derivatives 228 16.17. Addition Theorems 228 16.18. Double Arguments 228 16.19. Half Arguments 228 16.20. Jacobi's ImaginaryTransformation 228 16.21. ComplexArguments 229 16.22. Leading Terms of the Series in Ascending Pwrs. of u 229 16.23. Series Expansion in Terms of the Nomeq 229 16.24. Integrals of the Twelve Jacobian Elliptic Functions 229 16.25. Notation for the Integrals of the Squares of the

Twelve Jacobian Elliptic Functions 230 16.26. Integrals in Terms of the Elliptic Integral of the Second Kind 230 16.27. Theta Functions; Expansions in Terms of the Nomeq 230 16.28. Relations Betweenthe Squares of the Theta Functions 230 16.29. Logarithmic Derivatives of the Theta Functions 230 16.30. Logarithms of Theta Functions of Sum and Dif f erence 231 16.31. Jacobi's Notation for Theta Functions 231 16.32 Calculation of Jacobi's Theta Function 0 (u/m) by

Use of the Arithmetic-Geometric Mean 231 16.33. Addition of Quarter-Periods to Jacobi's Eta and Theta Functions 231 16.34. Relation of Jacobi's Zeta Function to the Theta Functions 232 16.35. Calculation of Jacobi's Zeta Function Z (u/m) by Use of the

Arithmetic-Geometric Mean 232 16.36. Neville's Notation for Theta Functions 232 16.37. Expression as Infinite Products 233 16.38. Expression as Infinite Series 233

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17. ELLIPTIC INTEGRALS 234 L. M. Milne-Thomson

17.1. Definition of Elliptic Integrals 234 17.2. Canonical Forms 234 17.3. Complete Elliptic Integrals of the First and Second Kinds 235 17.4. Incomplete Elliptic Integrals of the First and Second Kinds 237 17.5. Landen's Transformation 242 17.6. TheProcessof theArithmetic-GeometricMean 243 17.7. Elliptic Integrals of the Third Kind 244

18. WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 246 Thomas H. Southard

18.1. Definitions, Symbolism, Restrictions and Conventions 246 18.2. Homogen. Relations, Reduction Formulas and Processes 248 18.3. Special Values and Relations 250 18.4. Addition and Multiplication Formulas 252 18.5. Series Expansions 252 18.6. Derivatives and Differential Equations 257 18.7. Integrals 258 18.8. Conformal Mapping 259 18.9. Relations with Complete Elliptic Integrals K and K' and Their

Parameter m and with Jacobi's Elliptic Functions 266 18.10. Relations with Theta Functions 267 18.11. Expressing anyEll ipt . Function in Terms of ^ and 0>'. 268 18.12. CaseA = 0 268 18.13. EquianharmonicCase(g2 = 0,g3 = 1) 269 18.14. LemniscaticCase(g2 = l ,g 3 = 0) 275 18.15 Pseudo-LemniscaticCase(g2 = - l , g 3 = 0) 279

19. PARABOLIC CYLINDER FUNCTIONS 281 J.C.P. Miller

19.1. The Parabolic Cylinder Functions, Introductory 2

TheEquation A f — ( -W+ a)y = 0 281 dxz *

19.2 to 19.6. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations 281

19.7. to 19.11. Asymptotic Expansions 284 19.12. to 19.15. Connections With Other Functions

2

TheEquation - i f + (4-x2— a)y = 0 286 dxz *

19.16. to 19.19. Power Series, Standard Solns., Wronskian and Other Relations, Integral Representations 287

19.20. to 19.24. Asymptotic Expansions 288 19.25. Connections With Other Functions 290 19.26. Zeros 291 19.27. Bessel Functions of Order ±1/4, ±3/4 as Parabolic

Cylinder Functions 292 20. MATHIEU FUNCTIONS 293

Gertrude Blanch 20.1. Mathieu's Equation 293 20.2. Determination ofCharacteristic Values 293 20.3. Floquet 's Theorem and Its Consequences 298 20.4. Other Solutions of Mathieu's Equation 301 20.5. Properties of Orthogonality and Normalization 303

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20.6. Solutions of Mathieu'sModified Equation for Integral v 303 20.7. Representations by Integrals and Some Integral Equations 306 20.8. Other Properties 309 20.9. Asymptotic Representations 311 20.10. Comparative Notations 315 Table 20.1. Characteristic Values, Joining Factors,

Some Critical Values 316 Table 20.2. Coefficients Am andB m 318

21. SPHEROIDAL WAFE FUNCTIONS 319 Arnold N. Lowan

21.1. Definition of Elliptical Coordinates 319 21.2. Definition of Prolate Spheroidal Coordinates 319 21.3. Definition of Oblate Spheroidal Coordinates 319 21.4. Laplacian in Spheroidal Coordinates 319 21.5. Wave Equation in Prolate and Oblate Spheroidal Coordinates 319 21.6. Differential Equations for Radial and Angular

Spheroidal Wave Functions 320 21.7. Prolate Angular Functions 320 21.8. Oblate Angular Functions 323 21.9. Radial Spheroidal Wave Functions 323 21.10. Joining Factors for Prolate Spheroidal Wave Functions 324 21.11. Notation 325 Table 21.1. Eigenvalues —Prolate and Oblate 326 22. ORTHOGONAL POLYNOMIALS 332

Urs. W. Hochstrasser 22.1. Definition of Orthogonal Polynomials 332 22.2. Orthogonality Relations 333 22.3. Explicit Expressions 334 22.4. Special Values 336 22.5. Interrelations 336 22.6. Differential Equations 340 22.7. Recurrence Relations 341 22.8. Differential Relations 342 22.9. Generating Functions 342 22.10. Integral Representations 343 22.11. Rodrigues Formula 344 22.12. SumFormulas 344 22.13. Integrals Involving Orthogonal Polynomials 344 22.14. Inequalities 345 22.15. Limit Relations 346 22.16. Zeros 346 22.17. OrthogonalPolynominalsof aDiscreteVariable 347 22.18. Use and Extension of the Tables 348 22.19. Least Square Approximations 349 22.20. EconomizationofSeries 350 Table 22.1. Coefficients for the Jacobi Polynomials Pn

(°'p)(x) 351 Table 22.2. Coefficients for the Ultraspherical Polynomials

Cw(x) and for x° in Terms ofC<g>(x) 352 Table 22.3. Coefficients for the Chebyshev Polynomials Tn(x)

and for xnin Terms ofTm(x) 353 Table 22.5. Coefficients for the Chebyshev Polynomials Un (x)

and for xn in Terms of Um(x) 353 Table 22.7. Coefficients for the Chebyshev Polynomials Cn (x)

and for xn in Terms of Cm(x) 354

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Table 22.8. Coefficients for the Chebyshev Polynomials Sn (x) and for xn in Terms ofSm(x) 354

Table 22.9. Coefficients for the Legendre Polynomials Pn (x) and for xn in Terms ofPm(x) 355

Table 22.10. Coefficients for the Laguerre Polynomials Ln (x) and for xn in Terms ofLm(x) 356

Table 22.12. Coefficients for the Hermite Polynomials Hn(x) and for xn in Terms ofHm(x) 357

23. BERNOULLI AND EULER POLYNOMIALS — RIEMANN ZETA FUNCTION 358 Emilie V. Haynsworth and Karl Goldberg

23.1. Bernoulli and Euler Polynomials and Euler-Maclaurin Formula 358 23.2. Riemann Zeta Functions and Other Sums of Recip. Powers 361 Table 23.1. Coeffs.ofthe Bernoulli and Euler Polynomials 363 Table 23.2. Bernoulli and Euler Numbers 364

24. COMBINATORIAL ANALYSIS 365 K. Goldberg, M. Newman and E. Haynsworth

24.1. Basic Numbers 365 24.1.1. Binomial Coefficients 365 24.1.2. Multinomial Coefficients 366 24.1.3. Stirling Numbers of the First Kind 367 24.1.4. Stirling Numbers of the Second Kind 367 24.2. Parti t ions 368 24.2.1. Unrestricted Parti t ions 368 24.2.2. Parti t ions IntoDistinct Parts 368 24.3. Number Theoretic Functions 369 24.3.1. The Möbius Function 369 24.3.2. The Euler Totient Function 369 24.3.3. Divisor Functions 370 24.3.4. Primitive Roots 370

25. NUMERICAL INTERPOLATION, DIFFERENTIATION AND INTEGRATION 371 Philip J. Davis and Ivan Polonsky

25.1. Differences 371 25.2. Interpolation 372 25.3. Differentiation 376 25.4. Integration 379 25.5. Ordinary Differential Equations 390 Table 25.2. n-Point Coefficients fork-th Order Differentiation 392 Table 25.3. n-Point Lagrangian Integration Coefficients (3 <n <10) 393 Table 25.4. Abscissas and Weight Factors for Gaussian Integration 394 Table 25.5. Abscissas for Equal Weight Chebyshev Integration 398 Table 25.6. Abscissas and Weight Factors for Lobatto Integration 398 Table 25.7. Abscissas and Weight Factors for Gaussian Integration

for Integrands with a Logarithmic Singularity 398 Table 25.8. Abscissas and Weight Factors for Gaussian Integration

of Moments 399 Table 25.9. Abscissas and Weight Factors for Laguerre Integration 401 Table 25.10. Abscissas and Weight Factors for Hermite Integration 402 Table 25.11. Coefficients for Fi lon 'sQuadrature Formula 402

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26. PROBABILITY FUNCTIONS 403 Marvin Zelen and Norman C. Severo

26.1. Probabili ty Functions: Def initiqns and Properties 403 26.2. Normal orGaussian Probabili ty Function 407 26.3. Bivariate Normal Probabili ty Function 412 26.4. Chi-Square Probabili ty Function 416 26.5. Incomplete Beta Function 420 26.6. F-(Variance-Ratio) Distribution Function 422 26.7. Student 's t-Distribution 424 26.8. Methodsof Generating Random Numbers and Their Applications 425 26.9. Use and Extension oftheTables 429 Table 26.1. Normal Probabili ty Function and Derivatives (0<x<5) 435 Table 26.7. Probabili ty Integral of ^-Distribution (0<x2< 10) 439

27. MISCELLANEOUS FUNCTIONS 442 Irene A. Stegun

27.1. Debye Functions 442 27.2. Planck's Radiation Function 443 27.3. Einstein Functions 443 27.4. Sievert Integral 445

27.5. fm(x)= I tme-'"-fdt and Related Integrals 445

27.6. f(*)=f" f^dt 447

27.7. Dilogarithm (Spence's Integral) j{x) = -$ll=\dt 448

27.8. Clausen's Integral and Related Summations 449 27.9. Vector-Addition Coefficients 450

Subject Index 455 Index of Notations 467 Notation — Greek Letters 469 Miscellaneous Notations 469