CONFLUENT HYPERGEOMETRIC FUNCTIONS BY L. J. SLATER, D.LIT., PH.D. Formerly Bateson Research Fellow Newnham College, Cambridge Institut fur theoretssche Physfk Technische Hochschule Darmstadt CAMBRIDGE AT THE UNIVERSITY i960 PRESS Sweaterft*.B ^ 8 9
CONTENTS PREFACE poge xi CHAPTER I DIFFERENTIAL EQUATIONS SATISFIED BY CONFLUENT HYPERGEOMETRIC FUNCTIONS 1.1 Introduction i I.I.I Generalized hypergeometric functions i 1.2 Two solutions of Kummer's equation 3 1.2.1 Two further solutions of Kummer's equation 4 1.3 The second form of solutions of Kummer's equation 5 1.4 Kummer's first theorem 6 1.5 The first logarithmic solutions when b is an integer 6 1.5.1 The second logarithmic solutions when b is an integer 8 1.6 Whittaker's normalized equation 9 1.7 An alternative solution for Whittaker's equation ' 10 1.7.1 The logarithmic solutions of Whittaker's equation when zm is an integer 11 1.8 Kummer's second theorem 12 1.8.1 Bessel functions as special cases of confluent hypergeometric functions 12 1.9 Relations between Kummer's functions and Whittaker's functions 13 CHAPTER 2 DIFFERENTIAL PROPERTIES 2.1 The differentiation of Kummer's function 15 2.1.1 The derivatives of U(a; b; x) 16 2.1.2 The Wronskians of Kummer's equation 17 2.2 Recurrence relations for ±F X [a; b; x] 19 2.2.1 Recurrence relations for U(a; b; x) 20 2.2.2 Continuation formulae for U(a; b; x) 21 2.3 Addition theorems for ji^fa; b; x] 21 2.3.1 Addition theorems for U(a; b; x) 22
vi CONTENTS 2.3.2 Multiplication theorems for xf-^a; b; x] page 22 2.3.3 Multiplication theorems for U(a; b; x) 23 2.4 The derivatives of M kim (x) 23 2.4.1 The derivatives of W km (x) 25 2.4.2 The Wronskians of Whittaker's equation 26 2.5 Recurrence relations for M k>m (x) 26 2.5.1 Recurrence relations for W km (x) 27 2.5.2 Continuation formulae for Whittaker's functions 28 2.6 Addition theorems for M km (x) 29 2.6.1 Addition theorems for W km (x) 29 2.6.2 Multiplication theorems for M kjm (x) 30 2.6.3 Multiplication theorems for W km (x) 30 2.7 Expansions in series of Bessel functions 30 2.7.1 An elementary proof of the 4F 3 [i] summation theorem ~ 31 2.7.2 Expansion of Kummer's function in terms of I n (x) 32 2.7.3 Some further expansions 32 CHAPTER 3 INTEGRAL PROPERTIES 3.1 Elementary integrals for Kummer's function 34 3.1.1 Barnes's integral for Kummer's function 35 3.1.2 "Barnes and Euler type integrals for U(a; b; x) 37 3.1.3 Pochhammer's contour integrals for Kummer's function 38 3.1.4 The Pochhammer integrals for U{a; b; x) 41 3.2 Elementary indefinite integrals 42 3.2.1 The Laplace transforms of ji^a; b; x] 43 3.2.2 The inverse Laplace transform 45 3.2.3 The Laplace transform of U(a; b; x) 46 3.3 Mellin transforms of ^[a; b; x\ 47 3.3.1 Mellin transforms of U{a\ b; x) 49 3.4 The Hankel transforms 49
CONTENTS Vll 3.5 Elementary integrals for the Whittaker functions page 50 3.5.1 Barnes type integrals for the Whittaker functions 51 3.5.2 Pochhammer contour integrals for the Whittaker functions 51 3.6 The Laplace transforms of the Whittaker functions 52 3.7 Integrals involving pairs of Kummer's functions 54 3.7.1 Integrals involving pairs of Whittaker functions 56 3.8 Some expansions in series 56 CHAPTER 4 ASYMPTOTIC EXPANSIONS 4.1 Introduction ' 58 4.1.1 The asymptotic expansions in x for Kummer's function 58 4.1.2 The asymptotic expansions in x for U(a; b; x) 60 4.1.3 The asymptotic expansions in x for Whittaker's functions ~ 61 4.2 Converging factors for Kummer's functions 61 4.2.1 Converging factors for Whittaker's functions. 65 4.3 Approximations when b is large 65 4.3.1 Approximations for Whittaker's functions when m is large 66 4.4 Bessel functions as limiting cases of Kummer functions 67 4.4.1 Approximations in terms of Bessel functions when a is large 68 4.4.2 Bessel functions as limiting cases of Whittaker functions 69 4.4.3 Approximations for Whittaker functions in terms of Bessel functions, when k is large 69 4.5 Approximations when a and x are real, \x> \b a 71 4.5.1 Approximations when \b a ~ \x 71 4.5.2 Approximations when \b a > \x 72 4.5.3 Whittaker functions when k and x are large 73 4.6 Olver's theorems 76 4.6.1 Asymptotic expansions when a is large 79 4.6.2 Asymptotic expansions when k and x are large 81 4.6.3 Asymptotic expansions when 4^ 4= x 83 4.6.4 Asymptotic expansions when \k = x 86
Vlll CONTENTS CHAPTER 5 RELATED DIFFERENTIAL EQUATIONS AND PARTICULAR CASES OF THE FUNCTIONS 5.1 General transforms of Kummer's equation p#g e 89 5.2 Kummer's second theorem and the connection with Bessel functions 92 5.3 The Coulomb wave equation 93 5.4 Further forms of Whittaker's equation 93 5.4.1 Watson's fourth-order equation 94 5.5 The Laguerre polynomials 95 5.6 The incomplete gamma functions - 96 5.7 Transformations of Kummer's equation when m = 2. 98 5.7.1 The Poiseuille functions 100 5.8 The Schrodinger equation 100 5.9 Kamke's equation 101 CHAPTER 6 DESCRIPTIVE PROPERTIES 6.1 The distribution of the zeros 102 6.1.1 The curves of zeros. 104 6.1.2 The zeros of U(a; b; x) 105 6.1.3~ Approximations to the zeros 106 6.2 Expansions for the zeros 107 6.3 Nesting processes no 6.4 Zeros in 'a' no 6.5 The zeros in ' b' 112 6.5.1 The tabulation of zeros in x 112 6.6 The numerical evaluation of Kummer's function 113 6.7 Exponential and oscillatory regions 118 6.7.1 The Sonine-Polya theorem 119 6.7.2 Graphing Kummer's function 120
CONTENTS IX REFERENCES page 121 Table of the smallest positive zeros of xi b o-i (0-1)2-5 Table of ^[a; b; x] over the range APPENDIX I [a; b; x] over the range a = 4-0(0-1) o-i, APPENDIX II a = I-O(O-I) i-o, b = o-i (o-i) i-o, x = o-i (o-i) io-o Table of JF-^a; b; 1] over the range APPENDIX III a 11-0(0-2)2-0, b = 4-0(0-2) i-o, x = 1 233 SYMBOLIC INDEX OF DEFINITIONS 244 GENERAL INDEX 245