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 _pF_q 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