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 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 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. 35 4.3. Circular Functions 37 4.4. Inverse Circular Functions 45 4.5. Hyperbolic Functions 49 4.6. InverseHyberbolicFunctions 52 5. EXPONENTIAL AND RELATED FUNCTIONS .... 56 Walter Gautschi and William F. Cahill 5.1. Exponential Integral 56 5.2. Sine and Cosine 59 Table 5.1. Sine, Cosine and Exponential Integrals (0

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 anyEllipt. 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,g3 = 0) 275 18.15 Pseudo-LemniscaticCase(g2 = -l,g3 = 0) 279 19. PARABOLIC CYLINDER FUNCTIONS 281 J.C.P. Miller 19.1. The Parabolic Cylinder Functions, Introductory 2 TheEquation Af — (-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 -if + (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 12 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 andBm 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 ( p) Table 22.1. Coefficients for the Jacobi Polynomials Pn °' (x) 351 Table 22.2. Coefficients for the Ultraspherical Polynomials Cw(x) and for x° in Terms ofC(x) 352 Table 22.3. Coefficients for the Tn(x) n and for x in Terms ofTm(x) 353 Table 22.5. Coefficients for the Chebyshev Polynomials Un (x) n and for x in Terms of Um(x) 353 Table 22.7. Coefficients for the Chebyshev Polynomials Cn (x) n and for x in Terms of Cm(x) 354 13

Table 22.8. Coefficients for the Chebyshev Polynomials Sn (x) n and for x in Terms ofSm(x) 354 Table 22.9. Coefficients for the Legendre Polynomials Pn (x) n and for x in Terms ofPm(x) 355 Table 22.10. Coefficients for the Laguerre Polynomials Ln (x) n and for x in Terms ofLm(x) 356 Table 22.12. Coefficients for the Hermite Polynomials Hn(x) n and for x in Terms ofHm(x) 357 23. BERNOULLI AND EULER POLYNOMIALS — 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. Partitions 368 24.2.1. Unrestricted Partitions 368 24.2.2. Partitions 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

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