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### Spie Press Book

Mathematical Techniques for Engineers and Scientists
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Book Description

As technology continues to move ahead, modern engineers and scientists are frequently faced with difficult mathematical problems that require an ever greater understanding of advanced concepts. Designed as a self-study text for practicing engineers and scientists, as well as a useful reference, the book takes the reader from ordinary differential equations to more sophisticated mathematics--Fourier analysis, vector and tensor analysis, complex variables, partial differential equations, and random processes. The emphasis is on the use of mathematical tools and techniques. The general exposition and choice of topics appeals to a wide audience of applied practitioners.

Book Details

Date Published: 22 April 2003
Pages: 820
ISBN: 9781628410723
Volume: PM118
Errata

Preface
Symbols and Notation
1 DIFFERENTIAL EQUATIONS
1.1 INTRODUCTION
1.2 CLASSIFICATIONS
1.2.1 Solutions
1.3 FIRST-ORDER EQUATIONS
1.3.1 Separation of Variables
1.3.2 Linear Equations
1.3.3 Initial Condition
1.3.4 Applications
1.4 SECOND-ORDER LINEAR EQUATIONS
1.4.1 Homogeneous Equations: Fundamental Solution Sets
1.4.2 Constant Coefficient Equations
1.4.3 Nonhomogeneous Equations Part I
1.4.4 Nonhomogeneous Equations Part II
1.4.5 Cauchy-Euler Equations
1.5 POWER SERIES METHOD
1.5.1 Review of Power Series
1.6 SOLUTIONS NEAR AN ORDINARY POINT
1.6.1 Ordinary and Singular Points
1.6.2 General Method for Ordinary Points
1.7 LEGENDRE EQUATION
1.7.1 Legendre Polynomials:
1.7.2 Legendre Functions of the Second Kind:
1.8 SOLUTIONS NEAR A SINGULAR POINT
1.8.1 Method of Frobenius
1.9 BESSEL'S EQUATION
1.9.1 The Gamma Function:
1.9.2 Bessel Functions of the First Kind:
1.9.3 Bessel Functions of the Second Kind:
1.9.4 Differential Equations Related to Bessel's Equation
EXERCISES
2 SPECIAL FUNCTIONS
2.1 INTRODUCTION
2.2 ENGINEERING FUNCTIONS
2.2.1 Step and Signum (Sign) Functions
2.2.2 Rectangle and Triangle Functions
2.2.3 Sinc and Gaussian Functions
2.2.4 Delta and Comb Functions
2.3 FUNCTIONS DEFINED BY INTEGRALS
2.3.1 Gamma Functions
2.3.2 Beta Funcion
2.3.3 Digamma and Polygamma Functions
2.3.4 Error Functions and Fresnel Integrals
2.4 ORTHOGONAL POLYNOMIALS
2.4.1 Legendre Polynomials
2.4.2 Hermite Polynomials
2.4.3 Laguerre Polynomials
2.4.4 Chebyshev Polynomials
2.5 FAMILY OF BESSEL FUNCTIONS
2.5.1 Standard Bessel Functions
2.5.2 Modified Bessel Functions
2.5.3 Other Bessel Functions
2.6 FAMILY OF HYPERGEOMETRIC-LIKE FUNCTIONS
2.6.1 Pochhammer Symbol
2.6.2 Hypergeometric Function of Gauss
2.6.3 Confluent Hypergeometric Functions:
2.6.4 Generalized Hypergeometric Functions:
2.6.5 Applications Involving Hypergeometric Functions
2.7 SUMMARY OF NOTATIONS FOR SPECIAL FUNCTIONS
EXERCISES
3 MATRIX METHODS AND LINEAR VECTOR SPACES
3.1 INTRODUCTION
3.2 BASIC MATRIX CONCEPTS AND OPERATIONS
3.2.1 Algebraic Properties
3.2.2 Determinants
3.3.3 Special Matrices
3.3 LINEAR SYSTEMS OF EQUATIONS
3.3.1 Matrix Eigenvalue Problems
3.3.2 Real Symmetric and Skew-Symmetric Matrices
3.4 LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS
3.4.1 Homogeneous Systems
3.4.2 Homogeneous Systems with Constant Coefficients
3.4.3 Stability of Linear Systems
3.4.4 Nonhomogeneous Systems
3.5 LINEAR VECTOR SPACES
3.5.1 Linear Independence and Basis Vectors
3.5.2 Inner Product Spaces
3.5.3 Orthonormal Basis and the Expansion Theorem
3.5.4 Hilbert Spaces
EXERCISES
4 VECTOR ANALYSIS
4.1 INTRODUCTION
4.2 CARTESIAN COORDINATES
4.2.1 Base Vectors
4.2.2 Products of Vectors
4.2.3 Vector Identities
4.2.4 Applications
4.3 TENSOR NOTATION
4.3.1 Einstein Summation Convention
4.3.2 Kronecker Delta and Permutation Symbol:
4.3.3 Products of Vectors and Identities
4.4 VECTOR FUNCTIONS OF ONE VARIABLE
4.4.1 Space Curves
4.4.2 Frenet-Serret Formulas
4.4.3 Velocity and Acceleration
4.4.4 Planar Motion in Polar Coordinates
4.5 SCALAR AND VECTOR FIELDS
4.5.2 Divergence:
4.5.3 Physical Interpretation of Divergence
4.5.4 Curl:
4.5.5 Vector Differential Operators: Tensor Notation
4.6 LINE AND SURFACE INTEGRALS
4.6.1 Line Integrals
4.6.2 Conservative Fields
4.6.3 Surface Integrals
4.7 INTEGRAL RELATIONS BETWEEN LINE, SURFACE AND VOLUME INTEGRALS
4.7.1 Green's Theorem in the Plane
4.7.2 Theory of Harmonic Functions
4.7.3 Divergence Theorem and Stokes' Theorem
4.8 ELECTROMAGNETIC THEORY
4.8.1 Maxwell's Equations
4.8.2 Poisson's Equation
4.8.3 Electromagnetic Wave Equation
EXERCISES
5 TENSOR ANALYSIS
5.1 INTRODUCTION
5.2 TENSOR NOTATION
5.2.1 Special Symbols
5.3 RECTILINEAR COORDINATES
5.3.1 Definition of Tensor
5.3.2 Tensor Operations
5.3.3 Symmetric and Skew-Symmetric Tensors
5.4 BASE VECTORS
5.4.1 Covariant Base Vectors
5.4.2 Contravariant Base Vectors: Reciprocal Basis
5.4.3 Metric Tensor
5.5 VECTOR ALGEBRA
5.5.1 Permutation Symbols in Rectilinear Coordinates
5.5.2 Dot Product
5.5.3 Cross Product and Mixed Triple Product
5.6 RELATIONS BETWEEN TENSOR COMPONENTS
5.6.1 Raising and Lowering Indices
5.6.2 Physical Components
5.7 REDUCTION OF TENSORS TO PRINCIPAL AXES
5.7.1 Two-Dimensional Case
5.7.2 Three-Dimensional Case
5.8 TENSOR CALCULUS: RECTILINEAR COORDINATES
5.9 CURVILINEAR COORDINATES
5.9.1 Differentials as Tensors
5.9.2 Tensor Fields and Base Vectors
5.9.3 Metric Tensors
5.10 TENSOR CALCULUS: CURVILINEAR COORDINATES
5.10.1 Christoffel Symbols
5.10.2 Covariant Derivative
5.10.3 Absolute Derivative
5.11 RIEMANN-CHRISTOFFEL CURVATURE TENSOR
5.12 APPLICATIONS
5.12.2 Dynamics of a Particle: Newton's Second Law
5.12.3 Dielectric Tensor of an Anisotropic Medium
EXERCISES
6 COMPLEX VARIABLES
6.1 INTRODUCTION
6.2 BASIC CONCEPTS
6.2.1 Geometric Interpretation: The Complex Plane
6.2.2 Polar Coordinate Representation
6.2.3 Euler Formulas
6.2.4 Powers and Roots of Complex Numbers
6.3 COMPLEX FUNCTIONS
6.3.1 Loci and Terminology
6.3.2 Functions as Mappings
6.3.3 Limits and Continuity
6.4 THE COMPLEX DERIVATIVE
6.4.1 Cauchy-Riemann Equations
6.4.2 Analytic Functions
6.4.3 Harmonic Functions
6.5 ELEMENTARY FUNCTIONS PART I
6.5.1 Complex Exponential Function
6.5.2 Trigonometric Functions
6.5.3 Hyperbolic Functions
6.6 ELEMENTARY FUNCTIONS PART II
6.6.1 Complex Logarithm
6.6.2 Complex Powers
6.6.3 Inverse Trigonometric and Hyperbolic Functions
6.7 MAPPINGS BY ELEMENTARY FUNCTIONS
6.7.1 Orthogonal Families
6.7.2 Simple Polynomials
6.7.3 Reciprocal Mapping
6.7.4 Bilinear Transformations
6.7.5 Conformal Mapping
EXERCISES
7 COMPLEX INTEGRATION, LAURENT SERIES, AND RESIDUES
7.1 INTRODUCTION
7.2 LINE INTEGRALS IN THE COMPLEX PLANE
7.2.1 Bounded Integrals
7.3 CAUCHY'S THEORY OF INTEGRATION
7.3.1 Deformation of Contours
7.3.2 Integrals Independent of Path
7.3.3 Cauchy's Integral Formula
7.3.4 Cauchy's Generalized Formula
7.3.5 Bounds on Analytic Functions
7.4 INFINITE SERIES
7.4.1 Sequences and Series of Constants
7.4.2 Power Series
7.4.3 Laurent Series
7.4.4 Zeros and Singularities
7.5 RESIDUE THEORY
7.5.1 Residues
7.6 EVALUATION OF REAL INTEGRALS PART I
7.6.1 Rational Functions of Cos and/or Sin
7.6.2 Improper Integrals of Rational Functions
7.6.3 Fourier Transform Integrals
7.7 EVALUATION OF REAL INTEGRALS PART II
7.8 HARMONIC FUNCTIONS REVISITED
7.8.1 Harmonic Functions in the Half-Plane
7.8.2 Harmonic Functions in Circular Domains
7.8.3 Invariance of Laplace's Equation
7.9 HEAT CONDUCTION
7.9.1 Steady-State Temperatures in the Plane
7.9.2 Conformal Mapping
7.10 TWO-DIMENSIONAL FLUID FLOW
7.10.1 Complex Potential
7.10.2 Source, Sink, and Doublet Flows
7.11 FLOW AROUND OBSTACLES
7.11.1 Circulation and Lift
7.11.2 Flow Around a Cylinder
EXERCISES
8 FOURIER SERIES, EIGENVALUE PROBLEMS, AND GREEN'S FUNCTION
8.1 INTRODUCTION
8.2 FOURIER TRIGONOMETRIC SERIES
8.2.1 Periodic Functions as Power Signals
8.2.2 Convergence of the Series
8.2.3 Even and Odd Functions: Cosine and Sine Series
8.2.4 Nonperiodic Functions: Extensions to Other Intervals
8.3 POWER SIGNALS: EXPONENTIAL FOURIER SERIES
8.3.1 Parseval's Theorem and the Power Spectrum
8.4 EIGENVALUE PROBLEMS AND ORTHOGONAL FUNCTIONS
8.4.1 Regular Sturm-Liouville Systems
8.4.2 Generalized Fourier Series
8.4.3 Periodic Sturm-Liouville Systems
8.4.4 Singular Sturm-Liouville Systems
8.5 GREEN'S FUNCTION
8.5.1 One-Sided Green's Function
8.5.2 Boundary Value Problems
8.5.3 Bilinear Formula
EXERCISES
9 FOURIER AND RELATED TRANSFORMS
9.1 INTRODUCTION
9.2 FOURIER INTEGRAL REPRESENTATION
9.2.1 Cosine and Sine Integral Representations
9.3 FOURIER TRANSFORMS IN MATHEMATICS
9.3.1 Fourier Cosine and Sine Transforms
9.4 FOURIER TRANSFORMS IN ENGINEERING
9.4.1 Energy Spectral Density Function
9.4.2 Table of Fourier Transforms
9.4.3 Generalized Fourier Transforms
9.5 PROPERTIES OF THE FOURIER TRANSFORM
9.5.1 Time and Frequency Shifting
9.5.2 Differentiation and Integration
9.5.3 Convolution Theorem
9.6 LINEAR SHIFT-INVARIANT SYSTEMS
9.7 HILBERT TRANSFORMS
9.7.1 Analytic Signal Representation
9.7.2 Kramers-Kronig Relations
9.7.3 Table of Transforms and Properties
9.8 TWO-DIMENSIONAL FOURIER TRANSFORMS
9.8.1 Linear Systems in Optics
9.8.2 Coherent Imaging Systems
9.9 FRACTIONAL FOURIER TRANSFORM
9.9.1 Application in Optics
9.10 WAVELETS
9.10.1 Haar Wavelets
9.10.2 Wavelet Transform
EXERCISES
10 LAPLACE, HANKEL, AND MELLIN TRANSFORMS
10.1 INTRODUCTION
10.2 LAPLACE TRANSFORM
10.2.1 Table of Transforms and Operational Properties
10.2.2 Inverse Transforms I
10.2.3 Inverse Transforms II
10.3 INITIAL VALUE PROBLEMS
10.3.1 Simple Electric Circuits
10.3.2 Impulse Response Function
10.3.3 Stability of Linear systems
10.4 HANKEL TRANSFORM
10.4.1 Operational Properties and Table of Transforms
10.5 MELLIN TRANSFORM
10.5.1 Operational Properties and Table of Transforms
10.5.2 Complex Variable Methods
10.6 APPLICATIONS INVOLVING THE MELLIN TRANSFORM
10.6.1 Products of Random Variables
10.6.2 Electromagnetic Wave Propagation
10.7 DISCRETE FOURIER TRANSFORM
10.7.1 Discrete Transform Pair
10.8 Z-TRANSFORM
10.8.1 Operational Properties
10.8.2 Difference Equations
10.9 WALSH TRANSFORM
10.9.1 Walsh Functions
10.9.2 Walsh Series and the Discrete Walsh Transform
EXERCISES
11 CALCULUS OF VARIATIONS
11.1 INTRODUCTION
11.2 FUNCTIONALS AND EXTREMALS
11.2.1 Euler-Lagrange Equation
11.2.2 Special Cases of the Euler-Lagrange Equation
11.3 SOME CLASSICAL VARIATIONAL PROBLEMS
11.3.1 Shortest Arc Connecting Two Points
11.3.2 Surface of Revolution with Minimum Area
11.3.3 Brachistochrone Problem
11.4 VARIATIONAL NOTATION
11.4.1 Natural Boundary Conditions
11.5 OTHER TYPES OF FUNCTIONALS
11.5.1 Functionals with Several Dependent Variables
11.5.2 Functionals with Higher-Order Derivatives
11.5.3 Functionals with Several Independent Variables
11.6 ISOPERIMETRIC PROBLEMS
11.6.1 Constraints and Lagrange Multipliers
11.6.2 Sturm-Liouville Problem
11.7 RAYLEIGH-RITZ APPROXIMATION METHOD
11.7.1 Eigenvalue Problems
11.8 HAMILTON'S PRINCIPLE
11.8.1 Generalized Coordinates and Lagrange's Equations
11.8.2 Linear Theory of Small Oscillations
11.9 STATIC EQUILIBRIUM OF DEFORMABLE BODIES
11.9.1 Deflections of an Elastic String
11.9.2 Deflections of an Elastic Beam
11.10 TWO-DIMENSIONAL VARIATIONAL PROBLEMS
11.10.1 Forced Vibrations of an Elastic String
11.10.2 Equilibrium of a Stretched Membrane
EXERCISES
12 PARTIAL DIFFERENTIAL EQUATIONS
12.1 INTRODUCTION
12.2 CLASSIFICATION OF SECOND-ORDER PDES
12.3 THE HEAT EQUATION
12.3.1 Homogeneous Boundary Conditions
12.3.2 Nonhomogeneous Boundary Conditions
12.3.3 Derivation of the Heat Equation
12.4 THE WAVE EQUATION
12.4. d'Alembert's Solution
12.5 THE EQUATION OF LAPLACE
12.5.1 Rectangular Domain
12.5.2 Circular Domain
12.5.3 Maximum-Minimum Principle
12.6 GENERALIZED FOURIER SERIES
12.6.1 Convective Heat Transfer at One Endpoint
12.6.2 Nonhomogeneous Heat Equation
12.6.3 Nonhomogeneous Wave Equation
12.7 APPLICATIONS INVOLVING BESSEL FUNCTIONS
12.7.1 Vibrating Membrane
12.7.2 Scattering of Plane Waves By a Circular Cylinder
12.8 TRANSFORM METHODS
12.8.1 Heat Conduction on an Infinite Domain: Fourier Transform
12.8.2 Heat Conduction on a Semi-Infinite Domain: Laplace Transform
12.8.3 Nonhomogeneous Wave Equation
12.8.4 Poisson Integral Formula for the Half-Plane
12.8.5 Axisymmetric Dirichlet Problem for a Half-Space: Hankel Transform
EXERCISES
13 PROBABILITY AND RANDOM VARIABLES
13.1 INTRODUCTION
13.2 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
13.2.1 Cumulative Distribution Function
13.2.2 Probability Density Function
13.2.3 Discrete Random Variables
13.3 EXAMPLES OF DENSITY FUNCTIONS
13.3.1 Gaussian (or Normal) Distribution
13.3.2 Uniform Distribution
13.3.3 Rayleigh Distribution
13.3.4 Gamma Distribution
13.4 EXPECTED VALUES
13.4.1 Higher-Order Moments
13.4.2 Characteristic Functions
13.5 CONDITIONAL PROBABILITY
13.5.1 Conditional CDF and PDF
13.5.2 Expected Values
13.6 FUNCTIONS OF ONE RANDOM VARIABLE
13.6.1 Continuous Random Variables
13.6.2 Expected Values
13.6.3 Characteristic Function Method
13.7 TWORANDOM VARIABLES
13.7.1 Joint Distribution and Density Functions
13.7.2 Marginal Density Functions
13.7.3 Conditional Distributions and Densities
13.7.4 Independent Random Variables
13.7.5 Expected Values
13.7.6 Moments and Correlation
13.7.7 Bivariate Gaussian Distribution
13.8 FUNCTIONS OF TWO OR MORE RANDOM VARIABLES
13.8.1 Sums of Two Random Variables
13.8.2 Rician Distribution
13.8.3 Products of Random Variables
13.8.4 Quotients of Random Variables
13.8.5 Two Functions of Two Random Variables
13.8.6 Sums of Several Random Variables
13.9 LIMIT DISTRIBUTIONS
13.9.1 Gaussian Density Function
13.9.2 Gamma Density Function
EXERCISES
14 RANDOM PROCESSES
14.1 INTRODUCTION
14.2 PROBABILISTIC DESCRIPTION OF RANDOM PROCESS
14.2.1 First- and Second-Order Statistics
14.2.2 Stationary Random Processes
14.3 AUTOCORRELATION AND AUTOCOVARIANCE FUNCTIONS
14.3.1 Time Averages and Ergodicity
14.3.2 Basic Properties
14.3.3 Structure Functions
14.4 CROSS-CORRELATION AND CROSS-COVARIANCE
14.4.1 Basic Properties
14.5 POWER SPECTRAL DENSITY FUNCTIONS
14.5.1 Riemann-Stieltjes Integral
14.6 TRANSFORMATIONS OF RANDOM PROCESSES
14.6.1 Memoryless Nonlinear Transformations
14.6.2 Linear Systems
14.6.3 Correlation and Spectral Density Functions for the Output of a Linear System
14.7 STATIONARY GAUSSIAN PROCESSES
14.7.1 Multivariate Gaussian Distributions
14.7.2 Detection Devices
14.7.3 Zero-Crossing Problem
EXERCISES
15 APPLICATIONS
15.1 INTRODUCTION
15.2 MECHANICAL VIBRATIONS AND ELECTRIC CIRCUITS
15.2.1 Forced Oscillations I
15.2.2 Damped Motions
15.2.3 Forced Oscillations II
15.2.4 Simple Electric Circuits
15.3 BUCKLING OF A LONG COLUMN
15.4 COMMUNICATION SYSTEMS
15.4.1 Frequency Modulated Signals
15.4.2 Nonlinear Devices
15.4.4 Threshold Detection
15.5 APPLICATIONS IN GEOMETRICAL OPTICS
15.5.1 Eikonal Equation
15.5.2 Frenel-Serret Formulas Revisited
15.5.3 The Heated Window
15.6 WAVE PROPAGATION IN FREE SPACE
15.6.1 Hankel Transform Method
15.6.2 Huygens-Fresnel Integral: Lowest-order Gaussian Mode
15.6.3 Hermite-Gaussian Modes
15.7 ABCD RAY MATRICES FOR PARAXIAL SYSTEMS
15.7.1 Generalized Huygens-Fresnel Integral
15.7.2 Gaussian Lens
15.7.3 Fourier-Transform Plane
15.8 ZERNIKE POLYNOMIALS
15.8.1 Application in Optics
15.8.2 Atmospheric Effects on Imaging Systems
15.8.3 Aperture Filter Functions
EXERCISES
REFERENCES

### Preface

Modern engineers and scientists are frequently faced with difficult mathematical problems to solve. As technology continues to move ahead, some of these problems will require a greater understanding of advanced mathematical concepts than ever before. Unfortunately, the mathematical training in many engineering and science undergraduate university programs ends with an introductory course in differential equations. Even in those engineering and science curriculums that require some mathematics beyond differential equations, the required advanced mathematics courses often do not make a clear

This mathematics book is designed as a self-study text for practicing engineers and scientists, and as a useful reference source to complement more comprehensive publications. In particular, the text might serve as a supplemental text for certain undergraduate or graduate mathematics courses designed primarily for engineers and/or scientists. It takes the reader from ordinary differential equations to more sophisticated mathematics Fourier analysis, vector and tensor analysis, complex variables, partial differential equations, and random processes. The assumed formal training of the reader is at the undergraduate or beginning graduate level with possible extended experience on the job. We present the exposition in a way that is intended to bridge the gap between the formal education of the practitioner and his/her experience. The emphasis in this text is on the use of mathematical tools and techniques. In that regard it should be useful to those who have little or no experience in the subjects, but should also provide a useful review for readers with some background in the various topics.

The text is composed of fifteen chapters, each of which is presented independently of other chapters as much as possible. Thus, the particular ordering of the chapters is not necessarily crucial to the user with few exceptions. We begin Chapter 1 with a review of ordinary differential equations, concentrating on second-order linear equations. Equations of this type arise in simple mechanical oscillating systems and in the analysis of electric circuits. Special functions such as the gamma function, orthogonal polynomials, Bessel functions, and hypergeometric functions are introduced in Chapter 2. Our presentation also includes useful engineering functions like the step function, rectangle function, and delta (impulse) function. An introduction to matrix methods and linear vector spaces is presented in Chapter 3, the ideas of which are used repeatedly throughout the text. Chapters 4 and 5 are devoted to vector and tensor analysis, respectively. Vectors are used in the study of electromagnetic theory and to describe the motion of an object moving through space. Tensors are useful in studies of elasticity, general continuum mechanics, and in describing various properties of anisotropic materials like crystals. In Chapters 6 and 7 we present a fairly detailed discussion of analytic functions of a complex variable. The Cauchy-Riemann equations are developed in Chapter 6 along with the mapping properties associated with analytic functions. The Laurent series representation of complex functions and the residue calculus presented in Chapter 7 are powerful tools that can be used in a variety of applications, such as the evaluation of nonelementary integrals associated with various integral transforms.

Fourier series and eigenvalue problems are discussed in Chapter 8, followed by an introduction to the Fourier transform in Chapter 9. Generally speaking, the Fourier series representation is useful in describing spectral properties of power signals, whereas the Fourier transform is used in the same fashion for energy signals. However, through the development of formal properties associated with the impulse function, the Fourier transform can also be used for power signals. Other integral transforms are discussed in Chapter 10 the Laplace transform associated with initial value problems, the Hankel transform for circularly symmetric functions, and the Mellin transform for more specialized applications. A brief discussion of discrete transforms ends this chapter. We present some of the classical problems associated with the calculus of variations in Chapter 11, including the famous brachistochrone problem which is similar to Fermat's principle for light. In Chapter 12 we give an introductory treatment of partial differential equations, concentrating primarily on the separation of variables method and transform methods applied to the heat equation, wave equation, and Laplace's equation. Basic probability theory is introduced in Chapter 13 followed by a similar treatment of random processes in Chapter 14. The theory of random processes is essential to the treatment of random noise as found, for example, in the study of statistical communication systems. Chapter 15 is a collection of applications that involve a number of the mathematical techniques introduced in the first fourteen chapters. Some additional applications are also presented throughout the text in the various chapters.

In addition to the classical mathematical topics mentioned above, we also include a cursory introduction to some more specialized areas of mathematics that are of growing interest to engineers and scientists. These other topics include fractional Fourier transform (Chapter 9), wavelets (Chapter 9), and the Walsh transform (Chapter 10).

Except for Chapter 15, each chapter is a condensed version of a subject ordinarily expanded to cover an entire textbook. Consequently, the material found here is necessarily less comprehensive, and also generally less formal (i.e., it is presented in somewhat of a tutorial style). We discuss the main ideas that we feel are essential to each chapter topic and try to relate the mathematical techniques to a variety of applications, many of which are commonly associated with electrical and optical engineering e.g., communications, imaging, radar, antennas, and optics, among others. Nonetheless, we believe the general exposition and choice of topics should appeal to a wide audience of applied practitioners. At the end of each chapter is a 'Suggested Reading" section which contains a brief list of textbooks that generally provide a deeper treatment of the mathematical concepts. A more comprehensive set of references is also provided at the end of the text to which the reader is directed throughout the text by numbers, e.g., (see [10]). To further aid the reader, a short exercise set (generally 20-40 problems in each set) is also included at the end of each chapter. Most of the exercise sets have answers provided directly after the given problem. In addition, we have included a Symbols and Notation page for easy reference to some of the acronyms and special symbols as well as a list of Special Function notation (at the end of Chapter 2).