Book:Arthur Erdélyi/Higher Transcendental Functions Volume II

Harry Bateman: Higher Transcendental Functions, Volume I

Published $1953$, Dover Publications

ISBN 0-486-44614-X.

Online mirrors

hosted by Caltech

Contents

FOREWARD

CHAPTER VII

BESSEL FUNCTIONS FIRST PART: THEORY

7.1. Introduction

7.2. Bessel's differential equation

7.2.1. Bessel functions of general order

7.2.3. Kelvin's function and related functions

7.2.4. Bessel functions of integer order

7.2.5. Modified Bessel functions of integer order

7.2.6. Spherical Bessel functions

7.2.7. Products of Bessel functions

7.2.8. Miscellaneous results

7.3. Integral representations

7.3.1. Bessel coefficients

7.3.2. Integral representations of the Poisson type

7.3.3. Representations by loop integrals

7.3.4. Shläfli's, Gubler's, Sonine's and related integrals

7.3.5. Sommerfeld's integrals

7.3.6. Barnes' integrals

7.3.7. Airy's integrals

7.4. Asymptotic expansions

7.4.1. Large variable

7.4.2. Large order

7.4.3. Transitional regions

7.4.4. Uniform asymptotic expansions

7.5. Related functions

7.5.1. Neumann's and related polynomials

7.5.2. Lommel's poylnomials

7.5.3. Anger-Weber functions

7.5.4. Struves' functions

7.5.5. Lommel's functions

7.5.6. Some other notations and related functions

7.6. Addition theorems

7.6.1. Gegenbauer's addition theorem

7.6.2. Graf's addition theorem

7.7. Integral formulas

7.7.1. Indefinite integrals

7.7.2. Finite integrals

7.7.3. Infinite integrals with exponential functions

7.7.4. The discontinuous integral of Weber and Schafheitlin

7.7.5. Sonine and Gegenbauer's integrals and generalizations

7.7.6. Macdonald's and Nicholson's formulas

7.7.7. Integrals with respect to order

7.8. Relations between Bessel and Legendre functions

7.9. Zeros of the Bessel functions

7.10. Series and integral representations of arbitrary functions

7.10.1. Neumann's series

7.10.2. Kapteyn series

7.10.3. Schlömilch series

7.10.4. Fourier-Bessel and Dini series

7.10.5. Integral representations of arbitrary functions

SECOND PART: FORMULAS

7.11. Elementary relations and miscellaneous formulas

7.12. Integral representations

7.13. Asymptotic expansions

7.13.1. Large variable

7.13.2. Large order

7.13.3. Transitional regions

7.13.4. Uniform asymptotic expansions

7.14. Integral formulas

7.14.1. Finite integrals

7.14.2. Infinite integrals

7.15. Series of Bessel functions

References