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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 SeriesWelcome 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 fieldguides@SPIE.org. John E. Greivenkamp, Series Editor College of Optical Sciences The University of Arizona The Field Guide SeriesField 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 FunctionsMost 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 ContentsGlossary 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 berp(x), beip(x) Kelvin’s functions Bi(x) Airy function of the second kind B(x, y) Beta function Bn Bernoulli numbers Bn(x) Bernoulli polynomials cnu Elliptic function comb(x) Comb function C(x) Fresnel cosine integral Ci(x) Cosine integral
Gegenbauer polynomial dnu Elliptic function DE Differential equation δ(x) 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 Ep(x) 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 pFq Generalized hypergeometric function Gaus(x) Gaussian function
Meijer G function γ Euler’s constant γ(a, x), Γ(a, x) Incomplete gamma functions Γ(x) Gamma function
Hankel spherical Bessel functions Hn(x) Hermite polynomial
Hankel functions Hp(x) Struve function in(x) Modified spherical Bessel function of the first kind Ip(x) Modified Bessel function of the first kind jn(x) Spherical Bessel function of the first kind Jip(x) Integral Bessel function Jp(x) Bessel function of the first kind Jp(x) Anger function kerp(x), keip(x) Kelvin’s functions kn(x) Modified spherical Bessel function of the second kind Kp(x) Modified Bessel function of the second kind li(x) Logarithmic integral Ln(x) Laguerre polynomial
Associated Laguerre polynomial Lp(x) Modified Struve function Λ(x) Triangle function Mk, m(x) Whittaker function of the first kind n! Factorial function Pn(x) Legendre polynomial
Associated Legendre function of the first kind
Jacobi polynomial Π(x) Rectangle function
Elliptic integral of the third kind Qn(x) Legendre function of the second kind
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 Tn(x) Chebyshev polynomial of the first kind U(x) Step function U(a; c; x) Confluent hypergeometric function of the second kind Un(x) Chebyshev polynomial of the second kind Wk, m(x) Whittaker function of the second kind yn(x) Spherical Bessel function of the second kind Yp(x) Bessel function of the second kind
Spherical harmonic ψ(x) Digamma (psi) function ψ(m)(x) Polygamma function ζ(x) Riemann zeta function
“…is asymptotic to” || Absolute value
Binomial coefficient |
CITATIONS
Bessel functions
Adaptive optics
Atmospheric optics
Fiber lasers
Geometrical optics
Laser applications
Spherical lenses
