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

620.001'51–dc22

2011002631

Published by

SPIE

P.O. Box 10

Bellingham, Washington 98227-0010 USA

Phone: +1.360.676.3290

Fax: +1.360.647.1445

Email: books@spie.org

Web: http://spie.org

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

*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.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these

*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 Guides***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.*

**Field Guide**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

*topics as well as what material should be added to an individual volume to make these*

**Field Guide***more useful to you. Please contact us at fieldguides@SPIE.org.*

**Field Guides**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

*by L. C. Andrews and R. L. Phillips. Both books are SPIE Press publications.*

**Mathematical Techniques for Engineers and Scientists**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**

#### Glossary

(* a*)

_{n}

Pochhammer symbol

Ai(* x*)

Airy function of the first kind

ber_{p}(* x*), bei

_{p}(

*)*

**x**Kelvin’s functions

Bi(* x*)

Airy function of the second kind

* B*(

*,*

**x***)*

**y**Beta function

**B _{n}**

Bernoulli numbers

* B_{n}*(

*)*

**x**Bernoulli polynomials

cn**u**

Elliptic function

comb(* x*)

Comb function

* C*(

*)*

**x**Fresnel cosine integral

Ci(* x*)

Cosine integral

${C}_{n}^{\text{\lambda}}(x)$

Gegenbauer polynomial

dn**u**

Elliptic function

DE

Differential equation

δ(* x*)

Delta or impulse function

erf(* x*), erfc(

*)*

**x**Error functions

* E*(

**a**_{1}, …,

*;*

**a**_{p}

**c**_{1}, …,

*;*

**c**_{q}*)*

**x**MacRobert * E* function

$E(m,\text{\varphi})$

Elliptic integral of the second kind

Ei(* x*),

**E**_{1}(

*)*

**x**Exponential integrals

**E***_{p}*(

*)*

**x**Weber function

$F(m,\text{\varphi})$

Elliptic integral of the first kind

_{1}**F**_{1}(* a*;

*;*

**c***)*

**x**Confluent hypergeometric function of the first kind

_{2}**F**_{1}(* a*,

*;*

**b***;*

**c***)*

**x**Hypergeometric function

_{p}**F**_{q}

Generalized hypergeometric function

Gaus(* x*)

Gaussian function

${G}_{p,q}^{m,n}\left(x|\begin{array}{c}{a}_{p}\\ {c}_{q}\end{array}\right)$

Meijer * G* function

γ

Euler’s constant

γ(* a*,

*), Γ(*

**x***,*

**a***)*

**x**Incomplete gamma functions

Γ(* x*)

Gamma function

${h}_{p}^{(1)}(x),{h}_{p}^{(2)}(x)$

Hankel spherical Bessel functions

* H_{n}*(

*)*

**x**Hermite polynomial

${H}_{p}^{(1)}(x),{H}_{p}^{(2)}(x)$

Hankel functions

**H***_{p}*(

*)*

**x**Struve function

* i_{n}*(

*)*

**x**Modified spherical Bessel function of the first kind

* I_{p}*(

*)*

**x**Modified Bessel function of the first kind

* j_{n}*(

*)*

**x**Spherical Bessel function of the first kind

Ji_{p}(* x*)

Integral Bessel function

* J_{p}*(

*)*

**x**Bessel function of the first kind

**J***_{p}*(

*)*

**x**Anger function

ker_{p}(* x*), kei

_{p}(

*)*

**x**Kelvin’s functions

* k_{n}*(

*)*

**x**Modified spherical Bessel function of the second kind

* K_{p}*(

*)*

**x**Modified Bessel function of the second kind

li(* x*)

Logarithmic integral

* L_{n}*(

*)*

**x**Laguerre polynomial

${L}_{n}^{(a)}(x)$

Associated Laguerre polynomial

**L**_{p}(* x*)

Modified Struve function

Λ(* x*)

Triangle function

**M**_{k, m}(* x*)

Whittaker function of the first kind

* n*!

Factorial function

* P_{n}*(

*)*

**x**Legendre polynomial

${P}_{n}^{m}(x)$

Associated Legendre function of the first kind

${P}_{n}^{(a,b)}(x)$

Jacobi polynomial

Π(* x*)

Rectangle function

$\mathrm{\Pi}(m,\text{\varphi},a)$

Elliptic integral of the third kind

* Q_{n}*(

*)*

**x**Legendre function of the second kind

${Q}_{n}^{m}(x)$

Associated Legendre function of the second kind

rect(* x*)

Rectangle function

* S*(

*)*

**x**Fresnel sine integral

Si(* x*), si(

*)*

**x**Sine integral

sgn(* x*)

Signum (sign) function

sinc(* x*)

Sinc function

step(* x*)

Step function

tri(* x*)

Triangle function

* T_{n}*(

*)*

**x**Chebyshev polynomial of the first kind

* U*(

*)*

**x**Step function

* U*(

*;*

**a***;*

**c***)*

**x**Confluent hypergeometric function of the second kind

* U_{n}*(

*)*

**x**Chebyshev polynomial of the second kind

**W**_{k, m}(* x*)

Whittaker function of the second kind

* y_{n}*(

*)*

**x**Spherical Bessel function of the second kind

* Y_{p}*(

*)*

**x**Bessel function of the second kind

${Y}_{n}^{m}(\text{\theta},\text{\varphi})$

Spherical harmonic

ψ(* x*)

Digamma (psi) function

ψ^{(m)}(* x*)

Polygamma function

ζ(* x*)

Riemann zeta function

$\sim $

“…is asymptotic to”

||

Absolute value

$\left(\begin{array}{c}a\\ n\end{array}\right)$

Binomial coefficient