CONFLUENT HYPERGEOMETRIC FUNCTIONS

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

C A M B R I D G E

AT THE U N I V E R S I T Y PRESS

i960

CONTENTS

PREFACE pOge xi CHAPTER I D I F F E R E N T I A L E Q U A T I O N S S A T I S F I E D BY C O N F L U E N T H Y P E R G E O M E T R I C F U N C T I O N S 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

D I F F E R E N T I A L P R O P E R T I E S

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 ±FX[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 Mkim(x) 23

2.4.1 The derivatives of Wkm(x) 25

2.4.2 The Wronskians of Whittaker's equation 26 2.5 Recurrence relations for Mk>m(x) 26

2.5.1 Recurrence relations for Wkm(x) 27

2.5.2 Continuation formulae for Whittaker's functions 28

2.6 Addition theorems for Mkm(x) 29

2.6.1 Addition theorems for Wkm(x) 29

2.6.2 Multiplication theorems for Mkjm(x) 30

2.6.3 Multiplication theorems for Wkm(x) 30

2.7 Expansions in series of Bessel functions 30 2.7.1 An elementary proof of the 4F3[i] summation theorem ~ 31

2.7.2 Expansion of Kummer's function in terms of In(x) 32

2.7.3 Some further expansions 32

CHAPTER 3

I N T E G R A L P R O P E R T I E S

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 j i ^ 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

A S Y M P T O T I C E X P A N S I O N S

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 C O N T E N T S

CHAPTER 5

RELATED DIFFERENTIAL EQUATIONS AND PARTICULAR CASES OF THE FUNCTIONS

5.1 General transforms of Kummer's equation p#ge 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

D E S C R I P T I V E P R O P E R T I E S

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 n o 6.4 Zeros in 'a' n o 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

REFERENCES

Table of the smallest positive zeros of xi b — o-i (0-1)2-5

Table of ^[a; b; x] over the range

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

a — —11-0(0-2)2-0, b = —4-0(0-2) i-o, x = 1

SYMBOLIC INDEX OF DEFINITIONS

GENERAL INDEX

CONTENTS

APPENDIX I

[a; b; x] over the range a = —4-0(0-1) — o-i,

APPENDIX II A P P E N D I X I I I IX page 121 233 244 245