### Spie Press Book

Field Guide to Special Functions for EngineersFormat | Member Price | Non-Member Price |
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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.

- 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
to Other Functions_{p}F_{q} - 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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