SPECIAL FUNCTIONS
Contents
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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
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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
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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
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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
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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