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B.C. Carlson, ""

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 π

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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