Book:Arthur Erdélyi/Higher Transcendental Functions Volume II

Harry Bateman: Higher Transcendental Functions, Volume II

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

CHAPTER VIII FUNCTIONS OF THE PARABOLIC CYLINDER AND OF THE PARABOLOID OF REVOLUTION

8.1. Introduction

PARABOLIC CYLINDER FUNCTIONS

8.2. Definitions and elementary properties

8.3. Integral representations and integrals

8.4. Asymptotic expansions

8.5 Representation of functions in terms of the $D_{\nu}(x)$

8.5.1. Series

8.5.2. Representation by integrals with respect to the parameter

8.6 Zeros and descriptive properties

FUNCTIONS OF THE PARABOLOID OF REVOLUTION

8.7 The solutions of a particular confluent hypergeometric equation

8.8 Integrals and series involving functions of the paraboloid of revolution

CHAPTER IX THE INCOMPLETE GAMMA FUNCTIONS AND RELATED FUNCTIONS

9.1. Introduction

THE INCOMPLETE GAMMA FUNCTIONS

9.2. Definitions and elementary properties

9.2.1. The case of integer $a$

9.3. Integral representations and integral formulas

9.4. Series

9.5. Asymptotic representations

9.6. Zeros and descriptive properties

SPECIAL INCOMPLETE GAMMA FUNCTIONS

9.7. The exponential and logarithmic integral

9.8. Sine and cosine integrals

9.9. The error functions

9.10. Fresenel integrals and generalizations

References

CHAPTER X ORTHOGONAL POLYNOMIALS

10.1. Systems of orthogonal functions

10.2. The approximation problem

10.3. General properties of orthogonal polynomials

10.4. Mechanical quadrature

10.5. Continued fractions

10.6. The classical polynomials

10.7. General properties of the classical orthogonal polynomials

10.8. Jacobi polynomials

10.9. Gegenbauer polynomials

10.10. Legendre polynomials

10.11. Tchebichef polynomials

10.12. Laguerre polynomials

10.13. Hermite polynomials

10.14. Asymptotic behavior of Jacobi, Gegenbauer and Legendre polynomials

10.15. Zeros of Jacobi and related polynomials

10.16. Zeros of Laguerre and Hermite polynomials

10.17. Zeros of Laguerre and Hermite polynomials

10.18. Inequalities for the classical polynomials

10.19. Expansion problems

10.20. Examples of expansions

10.21. Some classes of orthogonal polynomials

10.22. Orthogonal polynomials of a discrete variable

10.23. Tchebichef's polynomials of a discrete variable and their generalizations

10.24. Krawtchouk's and related polynomials

10.25 Charlier's polynomials

References

See also

Book:Harry Bateman/Higher Transcendental Functions Volume I
Book:Harry Bateman/Higher Transcendental Functions Volume III