# Book:Larry C. Andrews/Special Functions of Mathematics for Engineers/Second Edition

## Larry C. Andrews: Special Functions of Mathematics for Engineers

Published $\text {1992}$, McGraw-Hill

ISBN 0-07-001848-0

### Contents

Preface to the Second Edition
Preface to the First Edition
Notation for Special Functions
Chapter 1. Infinite Series, Improper Integrals, and Infinite Products
1.1 Introduction
1.2 Infinite Series of Constants
1.2.1 The Geometric Series
1.2.2 Summary of Convergence Tests
1.2.3 Operations wllh Series
1.2.4 Factorials and Binomial Coefficients
1.3 Infinite Series of Functions
1.3.1 Properties of Uniformly Convergent Series
1.3.2 Power Series
1.3.3 Sums and Products of Power Series
1.4 Fourier Trigonometric Series
1.4.1 Cosine and Sine Series
1.5 Improper Integrals
1.5.1 Types of Improper Integrals
1.5.2 Convergence Tests
1.5.3 Pointwise and Uniform Convergence
1.6 Asymptotic Formulas
1.6.1 Small Arguments
1.6.2 Large Arguments
1.7 Infinite Products
1.7.1 Associated Infinite Series
1.7.2 Products of Functions
Chapter 2. The Gamma Function and Related Functions
2.1 Introduction
2.2 Gamma Function
2.2.1 Integral Representations
2.2.2 Legendre Duplication Formula
2.2.3 Weierstrass' Infinite Product
2.3 Applications
2.3.1 Miscellaneous Problems
2.3.2 Fractional-Order Derivatives
2.4 Beta Function
2.5 Incomplete Gamma Function
2.5.1 Asymptotic Series
2.6 Digamma and Polygamma Functions
2.6.1 Integral Representations
2.6.2 Asymptotic Series
2.6.3 Polygamma Functions
2.6.4 Riemann Zeta Function
Chapter 3. Other Functions Defined by Integrals
3.1 Introduction
3.2 Error Function and Related Functions
3.2.1 Asymptotic Series
3.2.2 Fresnel Integrals
3.3 Applications
3.3.1 Probability and Statistics
3.3.2 Heat Conduction In Solids
3.3.3 Vibrating Beams
3.4 Exponential Integral and Related Functions
3.4.1 Logarithmic Integral
3.4.2 Sine and Cosine Integrals
3.5 Elliptic Integrals
3.5.1 Limiting Values and Series Representations
3.5.2 The Pendulum Problem
Chapter 4. Legendre Polynomials and Related Functions
4.1 Introduction
4.2 Legendre Polynomials
4.2.1 The Generating Function
4.2.2 Special Values and Recurrence Formulas
4.2.3 Legendre's Differential Equation
4.3 Other Representations of the Legendre Polynomials
4.3.1 Rodrigues' Formula
4.3.2 Laplace Integral Formula
4.3.3 Some Bounds on $P_n(x)$
4.4 Legendre Series
4.4.1 Orthogonality of the Polynomials
4.4.2 Finite Legendre Series
4.4.3 Infinite Legendre Series
4.5 Convergence of the Series
4.5.1 Piecewise Continuous and Piecewise Smooth Functions
4.5.2 Pointwise Convergence
4.6 Legendre Functions of the Second Kind
4.6.1 Basic Properties
4.7 Associated Legendre Functions
4.7.1 Basic Properties of $P_n^m(x)$
4.8 Applications
4.8.1 Electric Potential due to a Sphere
4.8.2 Steady-State Temperatures In a Sphere
Chapter 5. Other Orthogonal Polynomials
5.1 Introduction
5.2 Hermite Polynomials
5.2.1 Recurrence Formulas
5.2.2 Hermite Series
5.2.3 Simple Harmonic Oscillator
5.3 Laguerre Polynomials
5.3.1 Recurrence Formulas
5.3.2 Laguerre Series
5.3.3 Associated Laguerre Polynomials
5.3.4 The Hydrogen Atom
5.4 Generalized Polynomial Sets
5.4.1 Gegenbauer Polynomials
5.4.2 Chebyshev Polynomials
5.4.3 Jacobi Polynomials
Chapter 6. Bessel Functions
6.1 Introduction
6.2 Bessel Functions of the First Kind
6.2.1 The Generating Function
6.2.2 Bessel Functions of Nonintegral Order
6.2.3 Recurrence Formulas
6.2.4 Bessel's Differential Equation
6.3 Integral Representations
6.3.1 Bessel's Problem
6.3.2 Geometric Problems
6.4 Integrals of Bessel Functions
6.4.1 Indefinite Integrals
6.4.2 Definite Integrals
6.5 Series Involving Bessel Functions
6.5.2 Orthogonality of Bessel Functions
6.5.3 Fourier-Bessel Series
6.6 Bessel Functions of the Second Klnd
6.6.1 Serles Expansion for $Y_n(x)$
6.6.2 Asymptotic Formulas for Small Arguments
6.6.3 Recurrence Formulas
6.7 Differential Equations Related to Bessel's Equation
6.7.1 The Oscillating Chaln
Chapter 7. Bessel Functions of Other Kinds
7.1 Introduction
7.2 Modified Bessel Functions
7.2.1 Modified Bessel Functions of the Second Kind
7.2.2 Recurrence Formulas
7.2.3 Generating Function and Addition Theorems
7.3 Integral Relations
7.3.1 Integral Representations
7.3.2 Integrals of Modified Bessel Functions
7.4 Spherical Bessel Functions
7.4.1 Recurrence Formulas
7.4.2 Modified Spherical Bessel Functions
7.5 Other Bessel Functions
7.5.1 Hankel Functions
7.5.2 Struve Functions
7.5.3 Kelvin's Functions
7.5.4 Airy Functions
7.6 Asymptotlc Formulas
7.6.1 Small Arguments
7.6.2 Large Arguments
Chapter 8. Applications Involving Bessel Functions
8.1 Introductions
8.2 Problems in Mechanics
8.2.1 The Lengthening Pendulum
8.2.2 Buckling of a Long Column
8.3 Statistical Communication Theory
8.3.1 Narrowband Nolse and Envelope Detection
8.4 Heat Conduction and Vibration Phenomena
8.4.1 Radial Symmetric Problems Involving Circles
8.4.2 Radial Symmetric Problems Involving Cylinders
8.4.3 The Helmholtz Equatlon
8.5 Step-Index Optical Fibers
Chapter 9. The Hypergeometric Function
9.1 Introduction
9.2 The Pochhammer Symbol
9.3 The Function $F(a, b; c; x)$
9.3.1 Elementary Properties
9.3.2 Integral Representation
9.3.3 The Hypergeometric Equation
9.4 Relation to Other Functions
9.4.1 Legendre Functions
9.5 Summing Series and Evaluating Integrals
9.5.1 Action-Angle Variables
Chapter 10. The Confluent Hypergeometric Functions
10.1 Introduction
10.2 The Functions $M(a; c; x)$ and $U(a; c; x)$
10.2.1 Elementary Properties of $M(a; c; x)$
10.2.2 Confluent Hypergeometric Equation and $U(a; c; x)$
10.2.3 Asymptotic Formulas
10.3 Relation to Other Functions
10.3.1 Hermite Functions
10.3.2 Laguerre Functions
10.4 Whittaker Functions
Chapter 11. Generalized Hypergeometric Functions
11.1 Introduction
11.2 The Set of Functions ${}_pF_q$
11.2.1 Hypergeometric-Type Series
11.3 Other Generalizations
11.3.1 The Meijer $G$ Function
11.3.2 The MacRobert $E$ Function
Chapter 12. Applications Involving Hypergeometric-Type Functions
12.1 Introduction
12.2 Statistical Communication Theory
12.2.1 Nonlinear Devices
12.3 Fluid Mechanics
12.3.1 Unsteady Hydrodynamic Flow Past an Infinite Plate
12.3.2 Transonic Flow and the Euler-Tricomi Equation
12.4 Random Fields
12.4.1 Structure Function of Temperature
Bibliography
Appendix: A List of Special Function Formulas