### Spie Press Book

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- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- 4.5.1 Gradient:
- 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
- SUGGESTED READING
- 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
- 5.8.1 Gradient, Divergence, and Curl
- 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
- 5.12.1 Gradient, Divergence, and Curl
- 5.12.2 Dynamics of a Particle: Newton's Second Law
- 5.12.3 Dielectric Tensor of an Anisotropic Medium
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- SUGGESTED READING
- 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
- 14.4.2 Cross-Correlation Techniques in Radar
- 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
- SUGGESTED READING
- 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
- 15.4.3 Coherent Detection Optical Receiver
- 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).

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