Book:Arthur Erdélyi/Higher Transcendental Functions Volume II
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Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions, Volume II
Published $1953$, Dover Publications
- ISBN 0-486-44614-X.
Online mirrors
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
- (1)
- (2)
- (3)
- (4)
- (5)
- (6)
- (7)
- (8)
- (9)
- (10)
- 7.2.2. Modified 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
- 7.2.1. Bessel functions of general order
- 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
- FIRST PART: THEORY
- 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
- 9.2. Definitions and elementary 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:Arthur Erdélyi/Higher Transcendental Functions Volume I
Book:Arthur Erdélyi/Higher Transcendental Functions Volume III
