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

Subject Matter

 * Special Functions

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
 * 6.5.1 Addition Formulas
 * 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
 * 8.3.2 Non-Rayleigh Radar Sea Clutter
 * 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
 * Selected Answers to Exercises


 * Index



Source work progress
* : $\S 1.3.2$: Power series: $(1.47)$