Special Functions

The following sections are included:

• FOREWORD

• PREFACE

• CONTENTS

In this chapter, we shall introduce to the readers some topics concerning the expansion of functions in infinite series and infinite products, and the asymptotic expansion of a function, which are usually not given in elementary courses in mathematics.

The following sections are included:

• Singular Points of Linear Ordinary Differential Equations of the Second Order

• Solution of the Equation in the Vicinity of an Ordinary Point

• Solution of the Equation in the Vicinity of a Singular Point

• Regular Solution. Regular Singularities

• Frobenius Method

• Point at Infinity

• Equations of Fuchsian Type

• Equations of Fuchsian Type Having Five Regular Singular Points

• Equations of Fuchsian Type Having Three Regular Singular Points

• Irregular Singularities. Formal Regular Solution

• Irregular Singularities. Normal Solutions and Subnormal Solutions

• Method of Solution by Integrals. Basic Principle

• Equations of Laplacian Type and Laplace Transform

• Euler Transform

The following sections are included:

• Definition of the Gamma Function

• Recurrence Relation

• The Infinite-Product Expression of Euler

• Weierstrass’ Infinite Product

• Relation between the Γ-Function and the Trigonometric Function

• Multiplication Formula

• Contour Integral

• Euler Integral of the First Kind. B-Function

• Double-Contour Integral

• Dirichlet Integral

• Logarithmic Derivative of the Γ-Function

• Asymptotic Expansions

• Another Derivation of the Asymptotic Expansion

• Riemann ς-Function

• The Functional Equation of the ς-Function

• The Value of ς(s,a) when s is an Integer

• Hermite Formula

• Relation to the Γ-Function

• Euler Product of the Γ-Function

• Riemann Integral of the ς-Function

• Another Derivation of the Asymptotic Expansion of the Γ-Function

• Evaluation of the ς-Function

Hypergeometric functions form an important class of special functions. Any Solution of the differential equation of Fuchsian type [Sec. 2.9] with 3 regular singular points can be expressed in terms of hypergeometric functions; for example, the Legendre functions, special spherical polynomials, Jacobi Polynomial, Chebyshev polynomial, etc…

The following sections are included:

• Legendre Equation

• Legendre Polynomials

• The Generating function of P_n(x). Rodrigues Formula

• Integral Representation of P_n(x)

• Recurrence Relations of P_n(x)

• Legendre Polynomials as a Complete Set of Orthonormal Functions

• Zeros of P_n(x)

• Legendre Functions of the Second Kind,Q_n(x)

• Recurrence Relations of Q_n(x)

• Expansion of the Function in Terms of Legendre Functions. Neumann Expansion

• Associate Legendre Functions

• Orthogonality Relations for

• Recurrence Relations for and

• Addition Formula

• Spherical Surface Harmonics Y_1m(θϕ

• The General Associate Legendre Functions

• Definition of on the cut: −∞<x<1

• Definition of on the cut: −∞<x<1

• Other Integral Expressions forP_v(z) and

• Addition Formulae

• Asymptotic Expansions of and when v→∞

• Ultra-Spherical Polynomials

• Excercise 5

Confluent hypergeometric functions are the solutions of the confluent hypergeometric equation. Such equation can be obtained from the hypergeometric equation by the confluence of two of its sigularities. Confluent hypergeometric functions include as special cases the commonly used Bessel functions, Hermite functions, Laguerre functions, etc. We shall discuss them form the view of solutions of a differential equation…

The following sections are included:

• Bessel Equation. Its Relation to the Confluent Hypergeometric Equation

• Bessel Functions of the First Kind: J_±v(z),2v≠ integer

• Bessel Functions of Order Half an Odd Integer: J_n+½(z)(n=0,±1±2…)

• Integral Representations of J_v(z)

• Bessel Functions of Integral Order J_n(z)(n=0,1,2,…)

• Bessel Functions of the Second Kind Y_v(z)

• Bessel Functions of the Third Kind (Hankel Functions)

• Modified (or, Imaginary Argument) Bessel Functions I_v(z) and K_v(z). Thomson Functions ber_v(z) and bei_v(z);etc …

• Spherical Bessel Functions

• Asymptotic Expansions for the case |z|→∞

• The Method of Steepest Descent

• Asymptotic Expansions of Bessel Functions of Order v for Large |v| and |z|

• Addition Formulae

• Integrals Containing Bessel Functions. (1) Finite Integrals

• Integrals Containing Bessel Functions. (1) Infinite Integrals

• Neumann Expansion

• Kapteyn Expansion

• The Zeros of Bessel Functions

• Fourier-Bessel Expansion

• Excercise 7

The following sections are included:

• Elliptic Integrals and Elliptic Functions

• The Periods of Elliptic Integrals

• The General Properties of Doubly-Periodic Functions and Elliptic Functions

• The Function ℘ (z)

• Algebraic Relation betweem ℘ (z)and ℘^′(z)

• The Function ς(z)

• The Function σ(z)

• Homogeneity of the Weierstrass Elliptic Function

• Representations of a General Elliptic Function

• Addition Formulae

• Expressing the Coordinates of a Cubic Curve by Elliptic Functions

• The Problem of a Quartic Polynomial

• Curves of Genus(Deficiency) 1

• Exercise 8

The following sections are included:

• The Theta Functions θ(v)

• The Functions ϑ_k(v)

• Elliptic Functions Represented by Theta Functions

• Relations among the Squares of ϑ_k(v)

• Addition Formulae

• Differential Equations Satisfied by Theta Functions

• The Values of Some Constants

• Legendre’s Elliptic Integral of the First Kind

• Jacobi’s Imaginary Transformation

• Transformation of Landen-Type

• Representation of Theta Functions by Infinite Product

• Fourier Expansion of the Logarithmic Derivatives of Theta Functions

• The Functions Θ(u) and H(u)

• Exercise 9

The following sections are included:

• Jacobian Elliptic Functions sn u,cn u, dn u

• Geometric Representations of Jacobian Elliptic Functions
  • The first geometric representation
  • The second geometric representation

• Complete Elliptic Integrals

• Addition Formulae

• The Periodicity of Jacobian Elliptic Functions

• The Zeros and Poles of Jacobian Elliptic Functions

• Transformations of Elliptic Functions

• Reductions of Elliptic Integrals

• Elliptic Integral of Second Kind

• Elliptic Integral of Third Kind

• Properties of the Function E(u)

• Differential Equations Satisfied by K and E with Respect to k and Expansions of K and E with Respect to k

• Relations between Jacobian Elliptic Functions and Theta Functions

• Expressing Jacobian Elliptic Functions in Infinite Products and Fourier Series

• Excercise 10

The following sections are included:

• Ellipsoidal Coordinates

• Representing the Coordinates wiht Elliptic Functions

• Lamé Equation

• Four Types of Lamé Functions

• Ellipsoidal Harmonics

• Niven’s Representation

• On the Zeros of Lamé Polynomials

• Generalized Lamé Functions

• Integral Equations of Lamé Functions

• The Integral Representation of Ellipsoidal Harmonics

• Exercise 11

The following sections are included:

• Mathieu Equation

• General Properties of the Solution. Fundamental Solutions

• Floquet Solution

• Perodic Solutions of Matheiu Equation

• Fourier Expansion of the Floquet Solution

• Formulae for Computing Eigenvalues λ(q)

• Mathieu Functions ce_m(z),m=0,1,2,… and se_m(z),m=1,2,…

• Expansion of λ_v(q) in Powers of q

• Fourier Expansions of ce_m(z) and se_m (z) for Small q

• Infinite Determinant

• Hill Equation

• Stable and Unstable Solutions of Mathieu Equation. Stable Region and Unstable Region

• Approximate Solutions of Mathieu Equation for λ > > q > 0

• Integral Equations for Mathieu Functions

• Exercise 12

The following sections are included:

• APPENDICES
  • Appendix I. Roots of a Cubic Equation
  • Appendix II. Roots of the Quartic Equation
  • Appendix III. Orthogonal Curvilinear Coordinate Systems

• BIBLIOGRAPHY

• GLOSSARY

• INDEX