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| Preface | |
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| List of Symbols | |
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| Series Solution of Differential Equations | |
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| Method of Frobenius | |
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| Examples | |
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| Problems | |
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| Gamma and Beta Functions | |
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| Definitions | |
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| Properties of the beta and gamma functions | |
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| Definition of the gamma function for negative values of the argument | |
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| Examples | |
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| Problems | |
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| Legendre Polynomials and Functions | |
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| Legendre's equation and its solution | |
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| Generating function for the Legendre polynomials | |
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| Further expressions for the Legendre polynomials | |
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| Explicit expressions for and special values of the Legendre polynomials | |
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| Orthogonality properties of the Legendre polynomials | |
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| Legendre series | |
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| Relations between the Legendre polynomials and their derivatives; recurrence relations | |
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| Associated Legendre functions | |
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| Properties of the associated Legendre functions | |
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| Legendre functions of the second kind | |
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| Spherical harmonics | |
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| Graphs of the Legendre functions | |
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| Examples | |
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| Problems | |
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| Bessel Functions | |
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| Bessel's equation and its solutions; Bessel functions of the first and second kind | |
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| Generating function for the Bessel functions | |
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| Integral representations for Bessel functions | |
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| Recurrence relations | |
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| Hankel functions | |
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| Equations reducible to Bessel's equation | |
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| Modified Bessel functions | |
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| Recurrence relations for the modified Bessel functions | |
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| Integral representations for the modified Bessel functions | |
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| Kelvin's functions | |
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| Spherical Bessel functions | |
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| Behaviour of the Bessel functions for large and small values of the argument | |
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| Graphs of the Bessel functions | |
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| Orthonormality of the Bessel functions; Bessel series | |
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| Integrals involving Bessel functions | |
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| Examples | |
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| Problems | |
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| Hermite Polynomials | |
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| Hermite's equation and its solution | |
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| Generating function | |
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| Other expressions for the Hermite polynomials | |
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| Explicit expressions for, and special values of, the Hermite polynomials | |
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| Orthogonality properties of the Hermite polynomials | |
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| Relations between Hermite polynomials and their derivatives; recurrence relations | |
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| Weber-Hermite functions | |
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| Examples | |
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| Problems | |
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| Laguerre Polynomials | |
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| Laguerre's equation and its solution | |
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| Generating function | |
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| Alternative expression for the Laguerre polynomials | |
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| Explicit expressions for, and special values of, the Laguerre polynomials | |
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| Orthogonality properties of the Laguerre polynomials | |
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| Relations between Laguerre polynomials and their derivatives; recurrence relations | |
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| Associated Laguerre polynomials | |
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| Properties of the associated Laguerre polynomials | |
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| Notation | |
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| Examples | |
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| Problems | |
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| Chebyshev Polynomials | |
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| Definition of Chebyshev polynomials; Chebyshev's equation | |
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| Generating function | |
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| Orthogonality properties | |
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| Recurrence relations | |
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| Examples | |
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| Problems | |
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| Gegenbauer and Jacobi Polynomials | |
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| Gegenbauer polynomials | |
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| Jacobi polynomials | |
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| Examples | |
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| Problems | |
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| Hypergeometric Functions | |
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| Definition of hypergeometric functions | |
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| Properties of the hypergeometric function | |
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| Properties of the confluent hypergeometric function | |
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| Examples | |
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| Problem | |
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| Other Special Functions | |
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| Incomplete gamma functions | |
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| Exponential integral and related functions | |
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| The error function and related functions | |
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| Riemann's zeta function | |
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| Debye functions | |
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| Elliptic integrals | |
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| Examples | |
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| Problems | |
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| Appendices | |
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| Convergence of Legendre series | |
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| Euler's constant | |
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| Differential equations | |
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| Orthogonality relations | |
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| Generating functions | |
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| Hints and Solutions to Problems | |
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| Bibliography | |
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| Index | |