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### 18. Special Functions and Their Properties

• 18.1. Some Coefficients, Symbols, and Numbers
• 18.1.1. Binomial Coefficients
• 18.1.2. Pochhammer Symbol
• 18.1.3. Bernoulli Numbers
• 18.1.4. Euler Numbers
• 18.2. Error Functions. Exponential and Logarithmic Integrals
• 18.2.1. Error Function and Complementary Error Function
• 18.2.2. Exponential Integral
• 18.2.3. Logarithmic Integral
• 18.3. Sine Integral and Cosine Integral. Fresnel Integrals
• 18.3.1. Sine Integral
• 18.3.2. Cosine Integral
• 18.3.3. Fresnel Integrals
• 18.4. Gamma Function, Psi Function, and Beta Function
• 18.4.1. Gamma Function
• 18.4.2. Psi Function (Digamma Function)
• 18.4.3. Beta Function
• 18.5. Incomplete Gamma and Beta Functions
• 18.5.1. Incomplete Gamma Function
• 18.5.2. Incomplete Beta Function
• 18.6. Bessel Functions (Cylindrical Functions)
• 18.6.1. Definitions and Basic Formulas
• 18.6.2. Integral Representations and Asymptotic Expansions
• 18.6.3. Zeros and Orthogonality Properties of Bessel Functions
• 18.6.4. Hankel Functions (Bessel Functions of the Third Kind)
• 18.7. Modified Bessel Functions
• 18.7.1. Definitions. Basic Formulas
• 18.7.2. Integral Representations and Asymptotic Expansions
• 18.8. Airy Functions
• 18.8.1. Definition and Basic Formulas
• 18.8.2. Power Series and Asymptotic Expansions
• 18.9. Degenerate Hypergeometric Functions (Kummer Functions)
• 18.9.1. Definitions and Basic Formulas
• 18.9.2. Integral Representations and Asymptotic Expansions
• 18.9.3. Whittaker Functions
• 18.10. Hypergeometric Functions
• 18.10.1. Various Representations of the Hypergeometric Function
• 18.10.2. Basic Properties
• 18.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions
• 18.11.1. Legendre Polynomials and Legendre Functions
• 18.11.2. Associated Legendre Functions with Integer Indices and Real Argument
• 18.11.3. Associated Legendre Functions. General Case
• 18.12. Parabolic Cylinder Functions
• 18.12.1. Definitions. Basic Formulas
• 18.12.2. Integral Representations, Asymptotic Expansions, and Linear Relations
• 18.13. Elliptic Integrals
• 18.13.1. Complete Elliptic Integrals
• 18.13.2. Incomplete Elliptic Integrals (Elliptic Integrals)
• 18.14. Elliptic Functions
• 18.14.1. Jacobi Elliptic Functions
• 18.14.2. Weierstrass Elliptic Function
• 18.15. Jacobi Theta Functions
• 18.15.1. Series Representation of the Jacobi Theta Functions. Simplest Properties
• 18.15.2. Various Relations and Formulas. Connection with Jacobi Elliptic Functions
• 18.16. Mathieu Functions and Modified Mathieu Functions
• 18.16.1. Mathieu Functions
• 18.16.2. Modified Mathieu Functions
• 18.17. Orthogonal Polynomials
• 18.17.1. Laguerre Polynomials and Generalized Laguerre Polynomials
• 18.17.2. Chebyshev Polynomials and Functions
• 18.17.3. Hermite Polynomials
• 18.17.4. Jacobi Polynomials and Gegenbauer Polynomials
• 18.18. Nonorthogonal Polynomials
• 18.18.1. Bernoulli Polynomials
• 18.18.2. Euler Polynomials
• References for Chapter 18

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