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This section contains the Introduction to the Series, the Series List, the Author's Introduction, the Table of Contents, and the Glossary of Symbols and Notation.

Library of Congress Cataloging-in-Publication Data

Andrews, Larry C.

Field guide to special functions for engineers / Larry C. Andrews.

p. cm. – (The field guide series; 18)

Includes bibliographical references and index.

ISBN 978-0-8194-8550-2

1. Engineering mathematics–Formulae–Handbooks, manuals, etc. I. Title.

TA332.A53 2011



Published by


P.O. Box 10

Bellingham, Washington 98227-0010 USA

Phone: +1.360.676.3290

Fax: +1.360.647.1445



The content of this book reflects the work and thought of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon.

Printed in the United States of America.


Introduction to the Series

Welcome to the SPIE Field Guides—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the SPIE Field Guides is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.

Each SPIE Field Guide addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.

The SPIE Field Guides are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at

John E. Greivenkamp, Series Editor

College of Optical Sciences

The University of Arizona

The Field Guide Series

Field Guide to Geometrical Optics, John E. Greivenkamp (FG01)

Field Guide to Atmospheric Optics, Larry C. Andrews (FG02)

Field Guide to Adaptive Optics, Robert K. Tyson & Benjamin W. Frazier (FG03)

Field Guide to Visual and Ophthalmic Optics, Jim Schwiegerling (FG04)

Field Guide to Polarization, Edward Collett (FG05)

Field Guide to Optical Lithography, Chris A. Mack (FG06)

Field Guide to Optical Thin Films, Ronald R. Willey (FG07)

Field Guide to Spectroscopy, David W. Ball (FG08)

Field Guide to Infrared Systems, Arnold Daniels (FG09)

Field Guide to Interferometric Optical Testing, Eric P. Goodwin & James C. Wyant (FG10)

Field Guide to Illumination, Angelo V. Arecchi; Tahar Messadi; R. John Koshel (FG11)

Field Guide to Lasers, Rüdiger Paschotta (FG12)

Field Guide to Microscopy, Tomasz S. Tkaczyk (FG13)

Field Guide to Laser Pulse Generation, Rüdiger Paschotta (FG14)

Field Guide to Infrared Systems, Detectors, and FPAs, Second Edition, Arnold Daniels (FG15)

Field Guide to Laser Fiber Technology, Rüdiger Paschotta (FG16)

Field Guide to Wave Optics, Dan Smith (FG17)

Field Guide to Special Functions for Engineers, Larry C. Andrews (FG18)

Field Guide to Special Functions

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 the functions’ basic properties. The functions treated in this 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. Useful engineering functions like the step function, rectangle function, and delta (impulse) function are also introduced.

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, University of Central Florida

Table of Contents

Glossary of Symbols and Notation x

Engineering Functions 1

Step and Signum (Sign) Functions 2

Rectangle and Triangle Functions 3

Sinc and Gaussian Functions 4

Delta Function 5

Delta Function Example 6

Comb Function 7

Infinite Series and Improper Integrals 8

Series of Constants 9

Operations with Series 10

Factorials and Binomial Coefficients 11

Factorials and Binomial Coefficients Example 12

Power Series 13

Operations with Power Series 14

Power Series Example 15

Improper Integrals 16

Asymptotic Series for Small Arguments 17

Asymptotic Series for Large Arguments 18

Asymptotic Series Example 19

Gamma Functions 20

Integral Representations of the Gamma Function 21

Gamma Function Identities 22

Incomplete Gamma Functions 23

Incomplete Gamma Function Identities 24

Gamma Function Example 25

Beta Function 26

Gamma and Beta Function Example 27

Digamma (Psi) and Polygamma Functions 28

Asymptotic Series 29

Bernoulli Numbers and Polynomials 30

Riemann Zeta Function 31

Other Functions Defined by Integrals 32

Error Functions 33

Fresnel Integrals 34

Exponential and Logarithmic Integrals 35

Sine and Cosine Integrals 36

Elliptic Integrals 37

Elliptic Functions 38

Cumulative Distribution Function Example 39

Orthogonal Polynomials 40

Legendre Polynomials 41

Legendre Polynomial Identities 42

Legendre Functions of the Second Kind 43

Associated Legendre Functions 44

Spherical Harmonics 45

Hermite Polynomials 46

Hermite Polynomial Identities 47

Hermite Polynomial Example 48

Laguerre Polynomials 49

Laguerre Polynomial Identities 50

Associated Laguerre Polynomials 51

Chebyshev Polynomials 52

Chebyshev Polynomial Identities 53

Gegenbauer Polynomials 54

Jacobi Polynomials 55

Bessel Functions 56

Bessel Functions of the First Kind 57

Properties of Bessel Functions of the First Kind 58

Bessel Functions of the Second Kind 59

Properties of Bessel Functions of the Second Kind 60

Modified Bessel Functions of the First Kind 61

Properties of the Modified Bessel Functions of the First Kind 62

Modified Bessel Functions of the Second Kind 63

Properties of the Modified Bessel Functions of the Second Kind 64

Spherical Bessel Functions 65

Properties of the Spherical Bessel Functions 66

Modified Spherical Bessel Functions 67

Hankel Functions 68

Struve Functions 69

Kelvin’s Functions 70

Airy Functions 71

Other Bessel Functions 72

Differential Equation Example 73

Bessel Function Example 74

Orthogonal Series 75

Fourier Trigonometric Series 76

Fourier Trigonometric Series: General Intervals 77

Exponential Fourier Series 78

Generalized Fourier Series 79

Fourier Series Example 80

Legendre Series 81

Hermite and Laguerre Series 82

Bessel Series 83

Bessel Series Example 84

Hypergeometric-Type Functions 85

Pochhammer Symbol 86

Hypergeometric Function 87

Hypergeometric Function Identities 88

Hypergeometric Function Example 89

Confluent Hypergeometric Functions 90

Confluent Hypergeometric Function Identities 91

Confluent Hypergeometric Function Example 92

Generalized Hypergeometric Functions 93

Relations of pFq to Other Functions 94

Meijer G Function 95

Properties of the Meijer G Function 96

Relation of the G Function to Other Functions 97

MacRobert E Function 98

Meijer G Example 99

Bibliography 101

Index 102



Pochhammer symbol


Airy function of the first kind

berp(x), beip(x)

Kelvin’s functions


Airy function of the second kind

B(x, y)

Beta function


Bernoulli numbers


Bernoulli polynomials


Elliptic function


Comb function


Fresnel cosine integral


Cosine integral


Gegenbauer polynomial


Elliptic function


Differential equation


Delta or impulse function

erf(x), erfc(x)

Error functions

E(a1, …, ap; c1, …, cq; x)

MacRobert E function


Elliptic integral of the second kind

Ei(x), E1(x)

Exponential integrals


Weber function


Elliptic integral of the first kind

1F1(a; c; x)

Confluent hypergeometric function of the first kind

2F1(a, b; c; x)

Hypergeometric function


Generalized hypergeometric function


Gaussian function


Meijer G function


Euler’s constant

γ(a, x), Γ(a, x)

Incomplete gamma functions


Gamma function


Hankel spherical Bessel functions


Hermite polynomial


Hankel functions


Struve function


Modified spherical Bessel function of the first kind


Modified Bessel function of the first kind


Spherical Bessel function of the first kind


Integral Bessel function


Bessel function of the first kind


Anger function

kerp(x), keip(x)

Kelvin’s functions


Modified spherical Bessel function of the second kind


Modified Bessel function of the second kind


Logarithmic integral


Laguerre polynomial


Associated Laguerre polynomial


Modified Struve function


Triangle function

Mk, m(x)

Whittaker function of the first kind


Factorial function


Legendre polynomial


Associated Legendre function of the first kind


Jacobi polynomial


Rectangle function


Elliptic integral of the third kind


Legendre function of the second kind


Associated Legendre function of the second kind


Rectangle function


Fresnel sine integral

Si(x), si(x)

Sine integral


Signum (sign) function


Sinc function


Step function


Triangle function


Chebyshev polynomial of the first kind


Step function

U(a; c; x)

Confluent hypergeometric function of the second kind


Chebyshev polynomial of the second kind

Wk, m(x)

Whittaker function of the second kind


Spherical Bessel function of the second kind


Bessel function of the second kind


Spherical harmonic


Digamma (psi) function


Polygamma function


Riemann zeta function

“…is asymptotic to”


Absolute value


Binomial coefficient


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