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POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged Edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A

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POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged Edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A 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. 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 9 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 /2 Table 14.2. C0fa) = e-"> ir(l + ir,)l 212 15. HYPERGEOMETRIC FUNCTIONS 213 Fritz Oberhettinger 15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument 213 10 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 = 0orl 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 11 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>'.
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