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CONTENTS xvii

18.6. Bessel Functions (Cylindrical Functions) . .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 947
18.6.1. Definitions and Basic Formulas .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 947
18.6.2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 949
18.6.3. Zeros and Orthogonality Properties of Bessel Functions . . . . . . . . . . . . . . . . . 951
18.6.4. Hankel Functions (Bessel Functions of the Third Kind) . . . . . . . . . . . . . . . . . . 952
18.7. Modified Bessel Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 953
18.7.1. Definitions. Basic Formulas .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 953
18.7.2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 954
18.8. Airy Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 955
18.8.1. Definition and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955
18.8.2. Power Series and Asymptotic Expansions .. .. .. .. .. ... .. .. .. .. .. .. .. .. 956
18.9. Degenerate Hypergeometric Functions (Kummer Functions) . . . . . . . . . . . . . . . . . . . . . 956
18.9.1. Definitions and Basic Formulas .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 956
18.9.2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 959
18.9.3. Whittaker Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960
18.10. Hypergeometric Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 960
18.10.1. Various Representations of the Hypergeometric Function . . . . . . . . . . . . . . 960
18.10.2. Basic Properties . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 960
18.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions . . . 962
18.11.1. Legendre Polynomials and Legendre Functions . . . . . . . . . . . . . . . . . . . . . . 962
18.11.2. Associated Legendre Functions with Integer Indices and Real Argument . . 964
18.11.3. Associated Legendre Functions. General Case .. .. ... .. .. .. .. .. .. .. .. 965
18.12. Parabolic Cylinder Functions .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 967
18.12.1. Definitions. Basic Formulas .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 967
18.12.2. Integral Representations, Asymptotic Expansions, and Linear Relations . . . 968
18.13. Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
18.13.1. Complete Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
18.13.2. Incomplete Elliptic Integrals (Elliptic Integrals) . . . . . . . . . . . . . . . . . . . . . . 970
18.14. Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972
18.14.1. Jacobi Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972
18.14.2. Weierstrass Elliptic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976
18.15. Jacobi Theta Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 978
18.15.1. Series Representation of the Jacobi Theta Functions. Simplest Properties . . 978
18.15.2. Various Relations and Formulas. Connection with Jacobi Elliptic Functions 978
18.16. Mathieu Functions and Modified Mathieu Functions . . . . . . . ... .. .. .. .. .. .. .. .. 980
18.16.1. Mathieu Functions .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 980
18.16.2. Modified Mathieu Functions .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 982
18.17. Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982
18.17.1. Laguerre Polynomials and Generalized Laguerre Polynomials . . . . . . . . . . . 982
18.17.2. Chebyshev Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983
18.17.3. Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985
18.17.4. Jacobi Polynomials and Gegenbauer Polynomials . . . . . . . . . . . . . . . . . . . . 986
18.18. Nonorthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988
18.18.1. Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988
18.18.2. Euler Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989
References for Chapter 18 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 990