Special Functions of Applied Mathematics - ISBN:0121601501

Special Functions of Applied Mathematics

Date: 22 April 2011, 13:25

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Maybe the best reference if you have to teach special functions.
Contents
1 Historical sketch
1.1 Introduction
1.2 The seventeenth century
1.3 The eighteenth century
'1.4 The nineteenth century
1.5 The twentieth century
Notes
2 Appell's symbol and hypergeometric series
2.1 Introduction,
2.2 Appell's symbol
2.3 Vandermonde's theorem
2.4 Hypergeometric series
2.5 Wallis' formula for ?
2.6 Arithmetic, geometric, and logarithmic means
2.7 Stirling's formula for n!
Notes
Formulary
Exercises
3 The gamma function
3.1 Introduction
3.2 Definition and difference equation
3.3 Analyticity
3.4 Limit formulas
3.5 Logarithmic convexity
3.6 The reciprocal of the gamma function
3.7 The duplication theorem
3.8 Stirling's formula in the complex plane
3.9 Euler's reflection formula
3.10 Inequalities
3 .11 Euler measures
Notes
Formulary
Exercises
4 The beta function
4.1 Introduction
4.2 The beta function of two variables
4.3 The beta function of several variables
4.4 Dirichlet measures
Notes
Formulary
Exercises
5 Dirichlet averages
5.1 Introduction
5.2 The averaging process
5.3 Averages of derivatives
5.4 Euler- Poisson differential equations
5.5 Newton-Taylor series with remainder
5.6 Associated functions
5.7 Averages of power series
5.8 Averages of ex
5.9 Averages of xt
5.10 Confluence
5.11 Averages of Cauchy's integral formula
5.12 Averages of e1/x
Notes
Formulary
Exercises
6 Averages of xn and xt
6.1 Introduction
6.2 Representation as a polynomial
6.3 Dirichlet averages with negative parameters
6.4 The binomial theorem
6.5 Linear transformation
6.6 Generating functions
6.7 Polynomials of Legendre, Gegenbauer, and Chebyshev
6.8 Analytic continuation of the R function
6.9 The first quadratic transformation
6.10 The second quadratic transformation
6.11 Gegenbauer's product formula
Notes
Formulary
Exercises
7 Jacobi polynomials
7.1 Taylor series and Jacobi series
7.2 Biorthogonality
7.3 The addition theorem for Gegenbauer polynomials
7.4 Asymptotic properties
7.5 Convergence of series
7.6 Jacobi series of an analytic function
7.7 Applications
7.8 Rodrigues' formula and orthogonality
7.9 Laguerre polynomials
7.10 Hermite polynomials
Notes
Exercises
8 Averages of xt
8.1 Evaluation of integrals
8.2 The Schwarz-Christoffel mapping and elliptic functions
8.3 Gauss's theorem and dependence on a small variable
8.4 Existence theorem for associated functions
8.5 Removal of integral parameters
Notes
Exercises
9 Elliptic Integrals
9.1 Introduction 257
9.2 Symmetric standard functions 261
9.3 Reduction to standard functions 266
9.4 Applications 270
9.5 Landen's transformation 275
9.6 The duplication theorem 278
9.7 The addition theorem 281
9.8 A reduction theorem 283
Notes 287
Exercises 287
A Notation for sets
B Integrals depending on a parameter
Solutions to exercises
References
Index

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