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Spie Press Book

Field Guide to Special Functions for Engineers
Author(s): Larry C. Andrews
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Book Description

This Field Guide is designed to provide engineers and scientists with a quick reference for special functions that are crucial to resolving modern engineering and physics problems. The functions treated in this book apply to many fields, including electro-optics, electromagnetic theory, wave propagation, heat conduction, quantum mechanics, probability theory, and electric circuit theory, among many other areas of application. A brief review of these important topics is included in this guide, as well as an introduction to some useful engineering functions such as the step function, rectangle function, and delta (impulse) function.


Book Details

Date Published: 27 June 2011
Pages: 116
ISBN: 9780819485502
Volume: FG18
Errata

Table of Contents
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Glossary of Symbols and Notation
Engineering Functions
Step and Signum (sign) Functions
Rectangle and Triangle Functions
Sinc and Gaussian Functions
Delta Function
Delta Function Example
Comb Function
Infinite Series and Improper Integrals
Series of Constants
Operations with Series
Factorials and Binomial Coefficients
Factorials and Binomial Coefficients Example
Power Series
Operations with Power Series
Power Series Example
Improper Integrals
Asymptotic Series for Small Arguments
Asymptotic Series for Large Arguments
Asymptotic Series Example
Gamma Functions
Integral Representations
Gamma Function Identities
Incomplete Gamma Functions
Incomplete Gamma Function Identities
Gamma Function Example
Beta Function
Gamma and Beta Example
Digamma (Psi) and Polygamma Functions
Asymptotic Series
Bernoulli Numbers and Polynomials
Riemann Zeta Function
Other Functions Defined by Integrals
Error Functions
Fresnel Integrals
Exponential and Logarithmic Integrals
Sine and Cosine Integrals
Elliptic Integrals
Elliptic Functions
Cumulative Distribution Function Example
Orthogonal Polynomials
Legendre Polynomials
Legendre Polynomial Identities
Legendre Functions of the Second Kind
Associated Legendre Functions
Hermite Polynomials
Hermite Polynomial Identities
Hermite Polynomial Example
Laguerre Polynomials
Laguerre Polynomial Identities
Associated Laguerre Polynomials
Chebyshev Polynomials
Chebyshev Polynomial Identities
Gegenbauer Polynomials
Jacobi Polynomials
Bessel Functions
Bessel Function of the First Kind
Properties of Bessel Functions of the First Kind
Bessel Function of the Second Kind
Properties of Bessel Functions of the Second Kind
Modified Bessel Function of the First Kind
Properties of Modified Bessel Functions of the First Kind
Modified Bessel Function of the Second Kind
Properties of Modified Bessel Functions of the Second Kind
Spherical Bessel Functions
Properties of Spherical Bessel Functions
Modified Spherical Bessel Functions
Hankel Functions
Struve Functions
Kelvin's Functions
Airy Functions
Other Related Bessel Functions
Differential Equation Example
Bessel Function Example
Orthogonal Series
Fourier Trigonometric Series
Fourier Trigonometric Series: General Intervals
Exponential Fourier Series
Generalized Fourier Series
Fourier Series Example
Legendre Series
Hermite and Laguerre Series
Bessel Series
Bessel Series Example
Hypergeometric-Type Functions
Pochhammer Symbol
Hypergeometric Function
Hypergeometric Function Identities
Confluent Hypergeometric Functions
Confluent Hypergeometric Function Identities
Generalized Hypergeometric Function
Hypergeometric Function Example
Confluent Hypergeometric Function Example
Relation of pFq to Other Functions
Meijer G Function
Properties of the Meijer G Function
Relation of the G Function to Other Functions
MacRobert E Function
Meijer G Example
Bibliography
Index

Preface

Most of the material chosen for this Field Guide is condensed from two textbooks: Special Functions of Mathematics for Engineers by L. C. Andrews and Mathematical Techniques for Engineers and Scientists by L. C. Andrews and R. L. Phillips. Both books are SPIE Press publications.

Many modern engineering and physics problems demand a thorough knowledge of mathematical techniques. In particular, it is important to recognize the various special functions (beyond the elementary functions) that may arise in practice as a solution to a differential equation or as a solution to some integral. It also helps to have a good understanding of their basic properties. The functions treated in this Field Guide are among the most important for engineers and scientists. They commonly occur in problems involving electro-optics, electromagnetic theory, wave propagation, heat conduction, quantum mechanics, probability theory, and electric circuit theory, among many other areas of application.

Because of the close association of power series and improper integrals with special functions, a brief review of these important topics is included in this guide. In addition, we also briefly introduce some of the useful engineering functions like the step function, rectangle function, and delta (impulse) function.

Unfortunately, notation for various engineering and special functions is not consistent among disciplines. Also, some special functions have more than one definition depending on the area of application. For these reasons, the reader is advised to be careful when using more than one reference source. The notation for the special functions adopted in this Field Guide is that which the author considers most widely used in practice.

Larry C. Andrews
Professor Emeritus, UCF


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