Difference between revisions of "Book:Arthur Erdélyi/Higher Transcendental Functions Volume II"

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{{Book|Higher Transcendental Functions, Volume I|1953|Dover Publications|0-486-44614-X|Harry Bateman}}
+
{{Book|Higher Transcendental Functions, Volume II|1953|Dover Publications|0-486-44614-X|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi}}
  
 
===Online mirrors===
 
===Online mirrors===
Line 8: Line 8:
 
===Contents===
 
===Contents===
 
:FOREWARD
 
:FOREWARD
−
:CHAPTER VII BESSEL FUNCTIONS FIRST PART: THEORY
+
:CHAPTER VII BESSEL FUNCTIONS
−
::7.1. Introduction
+
::FIRST PART: THEORY
−
::7.2. Bessel's differential equation
+
:::7.1. Introduction
−
:::7.2.1. Bessel functions of general order
+
:::7.2. Bessel's differential equation
−
:::7.2.3. Kelvin's function and related functions
+
::::7.2.1. Bessel functions of general order
−
:::7.2.4. Bessel functions of integer order
+
:::::(1)
−
:::7.2.5. Modified Bessel functions of integer order
+
:::::[[Bessel J|(2)]]
−
:::7.2.6. Spherical Bessel functions
+
:::::(3)
−
:::7.2.7. Products of Bessel functions
+
:::::(4)
−
:::7.2.8. Miscellaneous results
+
:::::(5)
−
::7.3. Integral representations
+
:::::(6)
−
:::7.3.1. Bessel coefficients
+
:::::(7)
−
:::7.3.2. Integral representations of the Poisson type
+
:::::(8)
−
:::7.3.3. Representations by loop integrals
+
:::::(9)
−
:::7.3.4. Shläfli's, Gubler's, Sonine's and related integrals
+
:::::(10)
−
:::7.3.5. Sommerfeld's integrals
+
::::7.2.2. Modified Bessel functions of general order
−
:::7.3.6. Barnes' integrals
+
::::7.2.3. Kelvin's function and related functions
−
:::7.3.7. Airy's integrals
+
:::::[[Kelvin ber|(19)]] (and [[Kelvin bei|(19)]])
−
::7.4. Asymptotic expansions
+
:::::[[Kelvin ker|(20)]] (and [[Kelvin kei|(20)]])
−
:::7.4.1. Large variable
+
::::7.2.4. Bessel functions of integer order
−
:::7.4.2. Large order
+
::::7.2.5. Modified Bessel functions of integer order
−
:::7.4.3. Transitional regions
+
::::7.2.6. Spherical Bessel functions
−
:::7.4.4. Uniform asymptotic expansions
+
::::7.2.7. Products of Bessel functions
−
::7.5. Related functions
+
::::7.2.8. Miscellaneous results
−
:::7.5.1. Neumann's and related polynomials
+
:::7.3. Integral representations
−
:::7.5.2. Lommel's poylnomials
+
::::7.3.1. Bessel coefficients
−
:::7.5.3. Anger-Weber functions
+
::::7.3.2. Integral representations of the Poisson type
−
:::7.5.4. Struves' functions
+
::::7.3.3. Representations by loop integrals
−
:::7.5.5. Lommel's functions
+
::::7.3.4. Shläfli's, Gubler's, Sonine's and related integrals
−
:::7.5.6. Some other notations and related functions
+
::::7.3.5. Sommerfeld's integrals
−
::7.6. Addition theorems
+
::::7.3.6. Barnes' integrals
−
:::7.6.1. Gegenbauer's addition theorem
+
::::7.3.7. Airy's integrals
−
:::7.6.2. Graf's addition theorem
+
:::7.4. Asymptotic expansions
−
::7.7. Integral formulas
+
::::7.4.1. Large variable
−
:::7.7.1. Indefinite integrals
+
::::7.4.2. Large order
−
:::7.7.2. Finite integrals
+
::::7.4.3. Transitional regions
−
:::7.7.3. Infinite integrals with exponential functions
+
::::7.4.4. Uniform asymptotic expansions
−
:::7.7.4. The discontinuous integral of Weber and Schafheitlin
+
:::7.5. Related functions
−
:::7.7.5. Sonine and Gegenbauer's integrals and generalizations
+
::::7.5.1. Neumann's and related polynomials
−
:::7.7.6. Macdonald's and Nicholson's formulas
+
::::7.5.2. Lommel's poylnomials
−
:::7.7.7. Integrals with respect to order
+
::::7.5.3. Anger-Weber functions
−
::7.8. Relations between Bessel and Legendre functions
+
::::7.5.4. Struves' functions
−
::7.9. Zeros of the Bessel functions
+
::::7.5.5. Lommel's functions
−
::7.10. Series and integral representations of arbitrary functions
+
::::7.5.6. Some other notations and related functions
−
:::7.10.1. Neumann's series
+
:::7.6. Addition theorems
−
:::7.10.2. Kapteyn series
+
::::7.6.1. Gegenbauer's addition theorem
−
:::7.10.3. Schlömilch series
+
::::7.6.2. Graf's addition theorem
−
:::7.10.4. Fourier-Bessel and Dini series
+
:::7.7. Integral formulas
−
:::7.10.5. Integral representations of arbitrary functions
+
::::7.7.1. Indefinite integrals
−
:SECOND PART: FORMULAS
+
::::7.7.2. Finite integrals
−
::7.11. Elementary relations and miscellaneous formulas
+
::::7.7.3. Infinite integrals with exponential functions
−
::7.12. Integral representations
+
::::7.7.4. The discontinuous integral of Weber and Schafheitlin
−
::7.13. Asymptotic expansions
+
::::7.7.5. Sonine and Gegenbauer's integrals and generalizations
−
:::7.13.1. Large variable
+
::::7.7.6. Macdonald's and Nicholson's formulas
−
:::7.13.2. Large order
+
::::7.7.7. Integrals with respect to order
−
:::7.13.3. Transitional regions
+
:::7.8. Relations between Bessel and Legendre functions
−
:::7.13.4. Uniform asymptotic expansions
+
:::7.9. Zeros of the Bessel functions
−
::7.14. Integral formulas
+
:::7.10. Series and integral representations of arbitrary functions
−
:::7.14.1. Finite integrals
+
::::7.10.1. Neumann's series
−
:::7.14.2. Infinite integrals
+
::::7.10.2. Kapteyn series
−
::7.15. Series of Bessel functions
+
::::7.10.3. Schlömilch series
−
::References
+
::::7.10.4. Fourier-Bessel and Dini series
−
[[Category:Books]]
+
::::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:Arthur Erdélyi/Higher Transcendental Functions Volume I]]<br />
 +
[[Book:Arthur Erdélyi/Higher Transcendental Functions Volume III]]<br />
 +
 
 +
[[Category:Book]]

Latest revision as of 05:44, 4 March 2018


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

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
(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
(19) (and (19))
(20) (and (20))
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:Arthur Erdélyi/Higher Transcendental Functions Volume I
Book:Arthur Erdélyi/Higher Transcendental Functions Volume III

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