CONFLUENT HYPERGEOMETRIC FUNCTIONS

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1 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

2 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 Two further solutions of Kummer's equation The second form of solutions of Kummer's equation Kummer's first theorem The first logarithmic solutions when b is an integer The second logarithmic solutions when b is an integer Whittaker's normalized equation An alternative solution for Whittaker's equation ' The logarithmic solutions of Whittaker's equation when zm is an integer Kummer's second theorem Bessel functions as special cases of confluent hypergeometric functions Relations between Kummer's functions and Whittaker's functions 13 CHAPTER 2 DIFFERENTIAL PROPERTIES 2.1 The differentiation of Kummer's function The derivatives of U(a; b; x) The Wronskians of Kummer's equation Recurrence relations for ±F X [a; b; x] Recurrence relations for U(a; b; x) Continuation formulae for U(a; b; x) Addition theorems for ji^fa; b; x] Addition theorems for U(a; b; x) 22

3 vi CONTENTS Multiplication theorems for xf-^a; b; x] page Multiplication theorems for U(a; b; x) The derivatives of M kim (x) The derivatives of W km (x) The Wronskians of Whittaker's equation Recurrence relations for M k>m (x) Recurrence relations for W km (x) Continuation formulae for Whittaker's functions Addition theorems for M km (x) Addition theorems for W km (x) Multiplication theorems for M kjm (x) Multiplication theorems for W km (x) Expansions in series of Bessel functions An elementary proof of the 4F 3 [i] summation theorem ~ Expansion of Kummer's function in terms of I n (x) Some further expansions 32 CHAPTER 3 INTEGRAL PROPERTIES 3.1 Elementary integrals for Kummer's function Barnes's integral for Kummer's function "Barnes and Euler type integrals for U(a; b; x) Pochhammer's contour integrals for Kummer's function The Pochhammer integrals for U{a; b; x) Elementary indefinite integrals The Laplace transforms of ji^a; b; x] The inverse Laplace transform The Laplace transform of U(a; b; x) Mellin transforms of ^[a; b; x\ Mellin transforms of U{a\ b; x) The Hankel transforms 49

4 CONTENTS Vll 3.5 Elementary integrals for the Whittaker functions page Barnes type integrals for the Whittaker functions Pochhammer contour integrals for the Whittaker functions The Laplace transforms of the Whittaker functions Integrals involving pairs of Kummer's functions Integrals involving pairs of Whittaker functions Some expansions in series 56 CHAPTER 4 ASYMPTOTIC EXPANSIONS 4.1 Introduction ' The asymptotic expansions in x for Kummer's function The asymptotic expansions in x for U(a; b; x) The asymptotic expansions in x for Whittaker's functions ~ Converging factors for Kummer's functions Converging factors for Whittaker's functions Approximations when b is large Approximations for Whittaker's functions when m is large Bessel functions as limiting cases of Kummer functions Approximations in terms of Bessel functions when a is large Bessel functions as limiting cases of Whittaker functions Approximations for Whittaker functions in terms of Bessel functions, when k is large Approximations when a and x are real, \x> \b a Approximations when \b a ~ \x Approximations when \b a > \x Whittaker functions when k and x are large Olver's theorems Asymptotic expansions when a is large Asymptotic expansions when k and x are large Asymptotic expansions when 4^ 4= x Asymptotic expansions when \k = x 86

5 Vlll CONTENTS CHAPTER 5 RELATED DIFFERENTIAL EQUATIONS AND PARTICULAR CASES OF THE FUNCTIONS 5.1 General transforms of Kummer's equation p#g e Kummer's second theorem and the connection with Bessel functions The Coulomb wave equation Further forms of Whittaker's equation Watson's fourth-order equation The Laguerre polynomials The incomplete gamma functions Transformations of Kummer's equation when m = The Poiseuille functions The Schrodinger equation Kamke's equation 101 CHAPTER 6 DESCRIPTIVE PROPERTIES 6.1 The distribution of the zeros The curves of zeros The zeros of U(a; b; x) ~ Approximations to the zeros Expansions for the zeros Nesting processes no 6.4 Zeros in 'a' no 6.5 The zeros in ' b' The tabulation of zeros in x The numerical evaluation of Kummer's function Exponential and oscillatory regions The Sonine-Polya theorem Graphing Kummer's function 120

6 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 = SYMBOLIC INDEX OF DEFINITIONS 244 GENERAL INDEX 245

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