### Spie Press Book

Special Functions for Optical Science and EngineeringFormat | Member Price | Non-Member Price |
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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

### 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
*T*(_{n}*x*) and*U*(_{n}*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
*Y*(θ,ϕ)^{m}_{l} - 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
*a*and_{n}*b*_{n} - 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
_{2}*F*_{1}and_{1}*F*_{1}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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