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### Spie Press Book

Special Functions for Optical Science and Engineering
Author(s): Vasudevan Lakshminarayanan; L. Srinivasa Varadharajan
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Book Description

This tutorial text is for those who use special functions in their work or study but are not mathematicians. Traditionally, special functions arise as solutions to certain linear second-order differential equations with variable coefficients—equations having applications in physics, chemistry, engineering, etc. This book introduces these differential equations, their solutions, and their applications in optical science and engineering. In addition to the common special functions, some less common functions are included. Also covered are Zernike polynomials, which are widely used in characterizing the quality of any imaging system, as well as certain integral transforms not usually covered in elementary texts. The book is liberally illustrated, and almost every chapter includes a set of Python 3.x codes that illustrate the use of these functions. Readers with a modest introduction to programming concepts will be able to modify these sample codes as needed.

"This is a monumental work that will be useful to people in many fields outside the optical domain. I have never seen all of this material in such a clear handbook-type format!"
--Alexander A. (Sandy) Sawchuk, Ph.D., University of Southern California

Book Details

Date Published: 23 December 2015
Pages: 408
ISBN: 9781628418873
Volume: TT103
Errata

Table of Contents
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### Table of Contents

Preface
Acknowledgment

1 Introduction
1.1 Some Famous Equations
1.2 Linearity
1.3 Maxwell's Equations
1.4 Curvilinear Coordinates
1.4.1 Spherical polar coordinates
1.4.2 Circular cylindrical coordinates
1.4.3 Elliptical cylindrical coordinates
1.5 Solution to the Helmholtz Equation
1.6 What's So Special About Special Functions?
1.7 Python Codes

2 Gamma, Beta, and Error Functions
2.1 Gamma Function
2.1.1 Factorial function
2.1.2 Stirling's approximation
2.1.3 Gamma function at some special points
2.1.4 The incomplete gamma function
2.1.5 Digamma function
2.2 Beta Function
2.2.1 Properties of the beta function
2.2.2 Applications of the beta function
2.3 Error Function
2.3.1 Error function and the normal distribution
2.3.2 Error function and the gamma function
2.3.3 Series expansion for the error function
2.3.4 Properties of the error function
2.3.5 Complementary error function

3 Other Integral Functions
3.1 Exponential Integrals
3.2 Logarithmic, Sine, and Cosine Integrals
3.3 Delta Function
3.3.1 Properties of the delta function
3.3.2 Application of the delta function
3.4 Kronecker Delta
3.5 Fresnel Integrals
3.6 Elliptic Integrals

4 Airy Functions
4.1 Introduction
4.2 Ai(x) and Bi(x) Functions
4.3 Relationship with Bessel Functions
4.4 Asymptotic Behavior of the Airy Function
4.5 Some Applications
4.5.1 Intensity near a caustic
4.5.2 Airy beams
4.5.3 Reflection of electromagnetic waves by the ionosphere
4.5.4 Schrödinger equation
4.5.5 Airy functions and the JWKB approximation

5 Bessel Functions
5.1 Introduction
5.2 Bessel Function of the First Kind
5.3 Bessel Functions of the Second Kind—Neumann Functions
5.4 Generalized Bessel Differential Equation
5.5 Bessel Functions of the Third Kind—Hankel Functions
5.6 Modified Bessel Functions
5.7 Kelvin Functions
5.8 Spherical Bessel Functions
5.9 Generating Function for Bessel Functions
5.10 Integral Relationship of Bessel Functions
5.11 Recurrence Relations of Bessel Functions
5.12 Approximate Formulas
5.13 Bessel–Fourier Series and Orthogonality
5.14 Examples
5.14.1 Modes of an optical waveguide
5.14.2 The Kaiser–Bessel window
5.14.3 Bessel beams
5.15 A Note on Python Coding
5.16 Summary

6 Chebyshev Polynomials
6.1 Introductions
6.2 Definition of Tn(x) and Un(x)
6.3 Recurrence Relations
6.4 Special Values
6.5 Generating Function
6.6 Orthogonality
6.7 Chebyshev Series
6.8 Applications

7 Hermite Polynomials
7.1 Introduction
7.2 Solution to the Hermite Equation
7.3 Series Solution
7.4 Generating Function
7.5 Recurrence Relations
7.6 Orthogonality
7.7 Examples
7.7.1 Quantum mechanical harmonic oscillator
7.7.2 Optical fiber modes with a quadratic index variation
7.7.3 Hermite–Gauss beams
7.7.4 Relativistic Hermite polynomials
7.7.5 Cortical receptive fields and vision
7.8 Summary

8 Gegenbauer, Jacobi, and Orthogonal Polynomials
8.1 Introduction
8.2 Gegenbauer Polynomials
8.2.1 Relationship to other orthogonal polynomials
8.3 Jacobi Polynomials
8.3.1 Relationship to Gegenbauer polynomials
8.3.2 Relationship to other polynomials
8.4 Classical Orthogonal Polynomial Functions and Differential Equation
8.4.1 Common properties
8.4.2 Alternative forms of the general differential equation
8.5 Summary

9 Laguerre Polynomials
9.1 Introduction
9.2 Series Solution
9.3 Generating Function and Recurrence Relations
9.4 Orthogonality
9.5 Integral Relationships
9.6 Associated Laguerre Polynomials
9.6.1 Generating function
9.6.2 Rodrigues's formula and other relations
9.7 A Warning
9.8 Examples
9.8.1 Optical fiber
9.8.2 Laguerre–Gauss beams
9.8.3 Data compression
9.9 Summary

10 Legendre Functions
10.1 Introduction
10.2 Series Solution
10.3 Generating Function and Recurrence Relations
10.3.1 Legendre polynomials in trigonometric form
10.4 Rodrigues's Formula
10.5 Integral Representation of Legendre Polynomials
10.6 Orthogonality
10.7 Associated Legendre Functions
10.8 Other Properties of the Associated Legendre Function
10.8.1 Orthogonality
10.8.2 Recurrence relations
10.8.3 Integral relationships
10.9 Spherical Harmonics
10.9.1 A note on (–1)m
10.9.2 Some properties of Yml (θ,ϕ)
10.9.3 Spherical harmonics addition theorem
10.10 Vector Spherical Harmonics
10.11 Examples
10.11.1 Multipole expansions
10.11.2 Geomagnetics
10.11.3 Computer graphics
10.12 Summary

11 Mathieu Functions
11.1 Introduction
11.2 Elliptical Coordinate System
11.3 Mathieu Differential Equation(s)
11.4 Angular Mathieu Function
11.4.1 Floquet's theorem
11.4.2 Hill equation
11.5 Series Solutions to the Mathieu Equation
11.6 Recurrence Relations and Other Factors
11.7 Evaluation of an and bn
11.8 Modified Mathieu Functions
11.9 List of Relationships and Identities
11.9.1 Relationship to Bessel functions
11.9.2 Modified Mathieu function identities
11.9.3 Asymptotic expansions of the radial Mathieu functions
11.9.4 Some derivative relationships
11.10 Nomenclature
11.11 A Note on Python Coding
11.12 Examples
11.12.1 Quantum pendulum
11.12.2 Mathieu beams
11.12.3 Nanoantennas
11.12.4 Paul traps
11.13 Summary

12 Hypergeometric Functions
12.1 Introduction
12.2 The Hypergeometric Function: Power Series Solution
12.3 Pochhammer Symbol
12.4 Indicial Equations for Hypergeometric Functions
12.4.1 Case 1
12.4.2 Case 2: c = 1
12.4.3 Case 3: c = 0, c = ±1, ±2, ±3, ±4, ...
12.5 Some Properties of Hypergeometric Functions
12.6 Solutions to the Hypergeometric Equation
12.7 The Confluent Hypergeometric Equation
12.8 Some Properties of the Confluent Hypergeometric Function
12.9 Relationship of 2F1 and 1F1 Functions to Other Functions
12.9.1 Relationship to elementary functions
12.9.2 Relationship to special functions
12.10 Asymptotic Expansions
12.11 Whittaker Functions
12.12 Summary

13 Integral Transforms
13.1 Introduction
13.1.1 Appearance of an integral transform
13.1.2 General integral transforms
13.1.3 Some properties of integral transforms
13.1.4 Summary
13.2 Hankel Transforms
13.2.1 Relationship to Fourier transform
13.2.2 Examples
13.3 Fresnel Transforms
13.3.1 Definitions and basic relationships
13.3.2 Fresnel zone plates
13.3.3 Comparison between the Dirac δ function, the Fourier transform, and the Fresnel transform
13.4 The Wigner Function
13.4.1 Definition
13.4.2 Properties of the Wigner function
13.4.3 Some examples of Wigner distribution functions
13.4.4 Two applications
13.5 Summary

14 Zernike Polynomials
14.1 Introduction
14.2 Description of Zernike Polynomials
14.3 Indexing Schemes
14.4 Python Codes for Zernike Polynomials
14.5 Integral Representation and Orthonormality of Zernike Polynomials
14.6 Recurrence Relations and Derivatives
14.7 Relationship to Other Special Functions
14.8 Relationship to Taylor Series and Seidel Aberrations
14.9 Primary Aberrations
14.10 Wavefront Error
14.11 Some Advantages of Using Zernike Polynomials
14.12 Conclusion

Appendix A: Series Solution of Differential Equations

Appendix B: Python Basics

Appendix C: Additional Reading

Postscript
References
Index

## Preface

According to a quote attributed to Albert Einstein,

"Formal symbolic representation of physical phenomena takes its rightful second place in a world where flowers and beautiful women abound."

That being said, the language of the optical scientist/engineer is mathematical! We only symbolically represent the beauty of optical phenomena.

It also happens that we primarily deal with second-order differential equations that describe these phenomena. In many cases of interest, these second-order differential equations take certain standard forms, usually depending on the coordinate system used. These standard forms have as solutions what are known as special functions. You are familiar with elementary functions such as trigonometric functions, exponential functions, etc. These are the second-stage Bessel functions, Hermite functions, and the like. In fact, special functions have even entered the cultural zeitgeist in an episode of the popular CBS sitcom, The Big Bang Theory, (season 5, episode 12 "The Bus Pants Utilization," where spherical Hankel functions are mentioned), where the main characters develop an app for smartphones that will solve differential equations in terms of special functions.

So, what do you need to get going? A basic course in calculus including, if possible, an introduction to linear differential equations (don't worry if you are not familiar with the Frobenius method; that is described in the appendix), a familiarity with solution by separation of variables, vectors, simple trigonometry, and basic ideas of complex numbers. You will also need some elementary knowledge of optics and electrodynamics, and some knowledge of quantum mechanics will be helpful. That's it! We have avoided complicated material such as complex variable theory, residue theorem, etc. If you are looking for rigid formalism, existence proofs, theorems, mathematical rigor and the like, you are out of luck with this book. You are better off going to another, more sophisticated text, some of which are listed in the Bibliography. We follow Richard Feynman's advice:

"However the emphasis should be somewhat more on how to do the mathematics quickly and easily and what formulas are true, rather than the mathematician's interest in methods of rigorous proof."

(He was commenting on operational calculus developed by Oliver Heaviside).

Now, this is not a traditional textbook. We have tried to explain the ideas as much as we can. However, as you know by now, the only way to master physics or math is to do problems. This book does not have exercises for you to do; the main reason for this is that there are many, many books that have umpteen unsolved problems for you. We wanted to write a "readable" book so that you can get a conceptual understanding, quickly. We have also tried to give examples from a wide range of optical science and engineering. We highly recommend that you consult the references and the bibliography for more information.

We have used Python code throughout the text. Python is a public domain scripting language that is quite easy to learn and is very powerful. If you are not familiar with it, Appendix B will give you a brief introduction. We assume that you are computer literate and that you are familiar with general concepts in programming. We encourage you to run the code(s) in the chapters. All codes provided in the book stick to the 'Minimum Working Example' types. You are encouraged to modify these and play with them to discover for yourself the properties of these special functions. This is an integral part of this book.

We hope you enjoy this book. We certainly did enjoy working on it. We appreciate your feedback so that if there is a second edition we can incorporate your suggestions.

Finally, it is said that a book does not get finished, it escapes from the authors. That is a truism in this case. There are many aspects and applications we would have/should have included; however, there are various constraints (time, book length, energy, to name a few) that have forced us to restrict the book to its current content. To our mind, the book is good. May you, the reader, learn and enjoy!

Vasudevan Lakshminarayan
L. Srinivasa Varadharajan

December 2015

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