POCKETBOOKOF MATHEMATICAL FUNCTIONS

Transcription

1 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

2 CONTENTS Forewordtothe 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. FactorsforConvertingCustomaryU.S. UnitstoSIUnits 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 Radioactivity and Radiation Protection ELEMENTARY ANALYTICAL METHODS 23 Milton Abramowitz 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometrie Progressions; Arithmetic, Geometrie, Harmonie and Generalized Means Inequalities Rules for Differentiation and Integration Limits, Maxima and Minima Absolute and Relative Errors Infinite Series Complex Numbers and Functions Algebraic Equations Successive Approximation Methods TheoremsonContinuedFractions ELEMENTARY TRANSCENDENTAL FUNCTIONS 33 Logarithmic, Exponential, Circular and Hyperbolic Functions Ruth Zucker 4.1. Logarithmic Function Exponential Function Circular Functions Inverse Circular Functions Hyperbolic Functions InverseHyberbolicFunctions EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS Walter Gautschi and William F. Cahill 5.1. Exponential Integral 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 E n (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

3 6. GAMMA FUNCTION AND RELATED FUNCTIONS 76 Philip J. Davis 6.1. Gamma Function Beta Function Psi(Digamma) Function Polygamma Functions Incomplete Gamma Function Incomplete Beta Function ERROR FUNCTION AND FRESNEL INTEGRALS 84 Walter Gautschi 7.1. Error Function Repeated Integrals ofthe Error Function Fresnel Integrals Definite and Indefinite Integrals 89 Table 7.7 Fresnel Integrals (0<x<5) LEGENDRE FUNCTIONS 94 Irene A. Stegun 8.1. Differential Equation Relations Between LegendreFunctions ValuesontheCut Explicit Expressions Recurrence Relations Special Values Trigonometrie Expansions Integral Representations Summation Formulas Asymptotic Expansions Toroidal Functions ConicalFunctions RelationtoEllipticIntegrals Integrals BESSEL FUNCTIONS OF INTEGER ORDER 102 F. W. J. Olver Bessel Functions J and Y Def initions and Elementary Properties Asymptotic Expansions for Large Arguments Asymptotic Expansions for Large Orders Polynomial Approximations Zeros 114 Modified Bessel Functions I and K Def initions and Properties Asymptotic Expansions Polynomial Approximations 122 Kelvin Functions Definitions and Properties Asymptotic Expansions 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

4 9 Table 9.9. ModifiedBesselFunctions Orders 3 9(0 <x<10) 148 Table Kelvin Functions Orders 0 and 1(0<x<5) 150 Kelvin Functions Modulus and Phase (0<x<7) BESSEL FUNCTIONS OF FRACTIONAL ORDER 154 H. A. Antosiewicz Spherical Bessel Functions Modified Spherical Bessel Functions Riccati-Bessel Functions Airy Functions 162 Table AiryFunctions(0<x<10) 169 Table Integrals of Airy Functions (0<x<10) 172 Table Zeros and Associated Values of Airy Functions and Their Derivatives (l<s<10) INTEGRALS OF BESSEL FUNCTIONS 173 Yudell L. Luke 11.1 Simple Integrals of Bessel Functions Repeated Integrals of J n (z) and K 0 (z) Reduction Formulas for Indefinite Integrals Definite Integrals 178 Table Integrals of Bessel Functions 182 Table Integrals of Bessel Functions STRUVE FUNCTIONS AND RELATED FUNCTIONS 185 Milton Abramowitz Struve Function H (z) Modified Struve Function L (z) Anger and Weber Functions 187 Table Struve Functions (0<x< o) CONFLUENT HYPERGEOMETRIC FUNCTIONS 189 Lucy Joan Slater Definitions of Kummer and Whittaker Functions Integral Representations Connections with Bessel Functions Recurrence Relations and Differential Properties Asymptotic Expansions and Limiting Forms Special Cases Zeros and Turning Values COULOMB WAVE FUNCTIONS 198 Milton Abramowitz Differential Equation, Series Expansions Recurrence and Wronskian Relations Integral Representations Bessel Function Expansions Asymptotic Expansions... ^ Special Values and Asymptotic Behavior _ Use and Extension of the Tables 203 Table Coulomb Wave Functions of Order Zero 204 Table C 0 fa) = e-"> /2 ir(l + ir,)l HYPERGEOMETRIC FUNCTIONS 213 Fritz Oberhettinger Gauss Series, Special Elementary Cases, Special Values of the Argument 213

5 Differentiation Formulas and Gauss' Relations for Contiguous Functions Integral Representations and Transformation Formulas Special casesof F(a, b;c;z),polynomials and Legendre Functions The Hypergeometric Differential Equation Riemann's Differential Equation AsymptoticExpansions i JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS 223 L. M. Milne-Thomson 16.1 Introduction Classification of the Twelve Jacobian Elliptic Functions Relation of the Jacobian Functions to the Copolar Trio Calculation of the Jacobian Functions by Use of the Arithmetic-GeometricMean(A. G. M.) Special Arguments Jacobian Functions whenm = 0orl Principal Terms Changeof Argument Relations Betweenthe Squares of the Functions Change of Parameter Reciprocal Parameter (Jacobi's Real Transformation) Descending Landen Transformation (Gauss' Transformation) Approximation in Terms of Circular Functions Ascending Landen Transformation Approximation in Terms of Hyperbolic Functions Derivatives Addition Theorems Double Arguments Half Arguments Jacobi's ImaginaryTransformation ComplexArguments Leading Terms of the Series in Ascending Pwrs. of u Series Expansion in Terms of the Nomeq Integrals of the Twelve Jacobian Elliptic Functions Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions Integrals in Terms of the Elliptic Integral of the Second Kind Theta Functions; Expansions in Terms of the Nomeq Relations Betweenthe Squares of the Theta Functions Logarithmic Derivatives of the Theta Functions Logarithms of Theta Functions of Sum and Dif f erence Jacobi's Notation for Theta Functions Calculation of Jacobi's Theta Function 0 (u/m) by Use of the Arithmetic-Geometric Mean Addition of Quarter-Periods to Jacobi's Eta and Theta Functions Relation of Jacobi's Zeta Function to the Theta Functions Calculation of Jacobi's Zeta Function Z (u/m) by Use of the Arithmetic-Geometric Mean Neville's Notation for Theta Functions Expression as Infinite Products Expression as Infinite Series 233

6 ELLIPTIC INTEGRALS 234 L. M. Milne-Thomson Definition of Elliptic Integrals Canonical Forms Complete Elliptic Integrals of the First and Second Kinds Incomplete Elliptic Integrals of the First and Second Kinds Landen's Transformation TheProcessof thearithmetic-geometricmean Elliptic Integrals of the Third Kind WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 246 Thomas H. Southard Definitions, Symbolism, Restrictions and Conventions Homogen. Relations, Reduction Formulas and Processes Special Values and Relations Addition and Multiplication Formulas Series Expansions Derivatives and Differential Equations Integrals Conformal Mapping Relations with Complete Elliptic Integrals K and K' and Their Parameter m and with Jacobi's Elliptic Functions Relations with Theta Functions Expressing anyellipt. Function in Terms of ^ and 0>' CaseA = EquianharmonicCase(g 2 = 0,g 3 = 1) LemniscaticCase(g 2 = l,g 3 = 0) Pseudo-LemniscaticCase(g 2 = -l,g 3 = 0) PARABOLIC CYLINDER FUNCTIONS 281 J.C.P. Miller The Parabolic Cylinder Functions, Introductory 2 TheEquation Af (-W+ a)y = dx z * 19.2 to Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations to Asymptotic Expansions to Connections With Other Functions 2 TheEquation -if + (4-x 2 a)y = dx z * to Power Series, Standard Solns., Wronskian and Other Relations, Integral Representations to Asymptotic Expansions Connections With Other Functions Zeros Bessel Functions of Order ±1/4, ±3/4 as Parabolic Cylinder Functions MATHIEU FUNCTIONS 293 Gertrude Blanch Mathieu's Equation Determination ofcharacteristic Values Floquet's Theorem and Its Consequences Other Solutions of Mathieu's Equation Properties of Orthogonality and Normalization 303

7 Solutions of Mathieu'sModified Equation for Integral v Representations by Integrals and Some Integral Equations Other Properties Asymptotic Representations Comparative Notations 315 Table Characteristic Values, Joining Factors, Some Critical Values 316 Table Coefficients A m andb m SPHEROIDAL WAFE FUNCTIONS 319 Arnold N. Lowan Definition of Elliptical Coordinates Definition of Prolate Spheroidal Coordinates Definition of Oblate Spheroidal Coordinates Laplacian in Spheroidal Coordinates Wave Equation in Prolate and Oblate Spheroidal Coordinates Differential Equations for Radial and Angular Spheroidal Wave Functions Prolate Angular Functions Oblate Angular Functions Radial Spheroidal Wave Functions Joining Factors for Prolate Spheroidal Wave Functions Notation 325 Table Eigenvalues Prolate and Oblate ORTHOGONAL POLYNOMIALS 332 Urs. W. Hochstrasser Definition of Orthogonal Polynomials Orthogonality Relations Explicit Expressions Special Values Interrelations Differential Equations Recurrence Relations Differential Relations Generating Functions Integral Representations Rodrigues Formula SumFormulas Integrals Involving Orthogonal Polynomials Inequalities Limit Relations Zeros OrthogonalPolynominalsof adiscretevariable Use and Extension of the Tables Least Square Approximations EconomizationofSeries 350 Table Coefficients for the Jacobi Polynomials P ( n ' p) (x) 351 Table Coefficients for the Ultraspherical Polynomials Cw(x) and for x in Terms ofc<g>(x) 352 Table Coefficients for the Chebyshev Polynomials Tn(x) and for x n in Terms oft m (x) 353 Table Coefficients for the Chebyshev Polynomials U n (x) and for x n in Terms of U m (x) 353 Table Coefficients for the Chebyshev Polynomials C n (x) and for x n in Terms of C m (x) 354

8 13 Table Coefficients for the Chebyshev Polynomials S n (x) and for x n in Terms ofs m (x) 354 Table Coefficients for the Legendre Polynomials P n (x) and for x n in Terms ofp m (x) 355 Table Coefficients for the Laguerre Polynomials L n (x) and for x n in Terms ofl m (x) 356 Table Coefficients for the Hermite Polynomials H n (x) and for x n in Terms ofh m (x) BERNOULLI AND EULER POLYNOMIALS RIEMANN ZETA FUNCTION 358 Emilie V. Haynsworth and Karl Goldberg Bernoulli and Euler Polynomials and Euler-Maclaurin Formula Riemann Zeta Functions and Other Sums of Recip. Powers 361 Table Coeffs.ofthe Bernoulli and Euler Polynomials 363 Table Bernoulli and Euler Numbers COMBINATORIAL ANALYSIS 365 K. Goldberg, M. Newman and E. Haynsworth Basic Numbers Binomial Coefficients Multinomial Coefficients Stirling Numbers of the First Kind Stirling Numbers of the Second Kind Partitions Unrestricted Partitions Partitions IntoDistinct Parts Number Theoretic Functions The Möbius Function The Euler Totient Function Divisor Functions Primitive Roots NUMERICAL INTERPOLATION, DIFFERENTIATION AND INTEGRATION 371 Philip J. Davis and Ivan Polonsky Differences Interpolation Differentiation Integration Ordinary Differential Equations 390 Table n-point Coefficients fork-th Order Differentiation 392 Table n-point Lagrangian Integration Coefficients (3 <n <10) 393 Table Abscissas and Weight Factors for Gaussian Integration 394 Table Abscissas for Equal Weight Chebyshev Integration 398 Table Abscissas and Weight Factors for Lobatto Integration 398 Table Abscissas and Weight Factors for Gaussian Integration for Integrands with a Logarithmic Singularity 398 Table Abscissas and Weight Factors for Gaussian Integration of Moments 399 Table Abscissas and Weight Factors for Laguerre Integration 401 Table Abscissas and Weight Factors for Hermite Integration 402 Table Coefficients for Filon'sQuadrature Formula 402

9 PROBABILITY FUNCTIONS 403 Marvin Zelen and Norman C. Severo Probability Functions: Def initiqns and Properties Normal orgaussian Probability Function Bivariate Normal Probability Function Chi-Square Probability Function Incomplete Beta Function F-(Variance-Ratio) Distribution Function Student's t-distribution Methodsof Generating Random Numbers and Their Applications Use and Extension ofthetables 429 Table Normal Probability Function and Derivatives (0<x<5) 435 Table Probability Integral of ^-Distribution (0<x 2 < 10) MISCELLANEOUS FUNCTIONS 442 Irene A. Stegun Debye Functions Planck's Radiation Function Einstein Functions Sievert Integral f m (x)= I t m e-'"-fdt and Related Integrals f(*)=f" f^dt Dilogarithm (Spence's Integral) dt 448 j{x) = -$ll=\ Clausen's Integral and Related Summations Vector-Addition Coefficients 450 Subject Index 455 Index of Notations 467 Notation Greek Letters 469 Miscellaneous Notations 469