SPECIAL FUNCTIONS

Contents


Chapter 1. Euler, Fourier, Bernoulli, Maclaurin, Stirling

1.1. The Integral Test and Euler's Constant...1
1.2. Fourier Series...2
1.3. Bernoulli Functions and Numbers...4
1.4. The Euler-Maclaurin Formulas...6
1.5. The Stirling Formulas...8

Chapter 2. The Gamma Function

2.1. Definition and Basic Properties...11
2.2. The Beta Function, Wallis' Product...13
2.3. The Reflection Formula...16
2.4. Stirling and Weierstrass...19
2.5. Evaluation of a Class of Infinite Products...21

Chapter 3. Elliptic Integrals and Elliptic Functions

3.1. Motivational Examples...23
3.2. General Definition of Elliptic Integrals...25
3.3. Evaluation of Elliptic Integrals...26
3.4. The Jacobian Elliptic Functions...29
3.5. Addition Theorems...30
3.6. Periodicity...31
3.7. Zeros, Poles, and Period Parallelograms...33
3.8. General Elliptic Functions...36
3.9. Weierstrass' P-Function...37
3.10. Elliptic Functions in Terms of P and P'...38
3.11. Elliptic Wheels - an Application...40
3.12. Miscellaneous Integrals...42

Chapter 4. Hypergeometric Functions

4.1. Solutions of Linear DEs at Regular Singular Points...43
4.2. Equations of Fuchsian Type...44
4.3. The Riemann-Papperitz Equation...46
4.4. The Hypergeometric Equation...48
4.5. Confluence of Singularities...51
4.6. Generalized Hypergeometric Functions...53

Chapter 5. Orthogonal Functions

5.1. Generating Functions...55
5.2. Orthogonality...57
5.3. Series Expansions...59

Copyright ©1995 by Leon M. Hall, all rights reserved
Leon Hall / lmhall@umr.edu / December 20, 1995