# Advanced Mathematical Methods in Science and Engineering

###### S. I. Hayek
Publisher:
Chapman & Hall/CRC
Publication Date:
2011
Number of Pages:
844
Format:
Hardcover
Edition:
2
Price:
129.95
ISBN:
9781420081978
Category:
Textbook
[Reviewed by
John D. Cook
, on
10/21/2010
]

S. I. Hayek’s Advanced Mathematical Methods in Science and Engineering covers a wide range of applied mathematics centered around differential equations. Half the chapters are explicitly devoted to differential equations: ODEs, series solutions to ODEs, boundary value problems, PDEs, and Green's functions. Most of the remaining chapters are closely related to computing solutions to differential equations: special functions, integral transforms, numerical methods, and asymptotic methods. The only chapter not directly tied to differential equations is an introduction to complex analysis. Even this chapter leads up to the Laplace transform inversion theorem which is necessary when using Laplace transfroms to compute solutions to differential equations.

The choice of topics in Advanced Mathematical Methods is typical of engineering mathematics handbooks with a few exceptions. There is more emphasis on special functions than usual, but less emphasis on linear algebra and numerical methods. Also, there is no mention of probability.

Hayek’s book contains a great deal of useful information in around 800 pages. It is a reference rather than a textbook, a compendium of results with little exposition to tie the results together. However, the chapters end with exercises — even some of the appendices have exercises — and solutions to the exercises are included in the back of the book.

The appearance of the book is disappointing. The figures do not have a consistent style and appear to have been pasted in from various sources. The mathematical typesetting is inconsistent, unattractive, and at times difficult to read.

In short, Advanced Mathematical Methods is more of a large collection of notes than a polished, cohesive book.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.

Ordinary Differential Equations
DEFINITIONS
LINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER
LINEAR INDEPENDENCE AND THE WRONSKIAN
LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF ORDER N WITH CONSTANT COEFFICIENTS
EULER’S EQUATION
PARTICULAR SOLUTIONS BY METHOD OF UNDETERMINED COEFFICIENTS
PARTICULAR SOLUTIONS BY THE METHOD OF VARIATIONS OF PARAMETERS
ABEL’S FORMULA FOR THE WRONSKIAN
INITIAL VALUE PROBLEMS

Series Solutions of Ordinary Differential Equations
INTRODUCTION
POWER SERIES SOLUTIONS
CLASSIFICATION OF SINGULARITIES
FROBENIUS SOLUTION

Special Functions
BESSEL FUNCTIONS
BESSEL FUNCTION OF ORDER ZERO
BESSEL FUNCTION OF AN INTEGER ORDER N
RECURRENCE RELATIONS FOR BESSEL FUNCTIONS
BESSEL FUNCTIONS OF HALF ORDERS
SPHERICAL BESSEL FUNCTIONS
HANKEL FUNCTIONS
MODIFIED BESSEL FUNCTIONS
GENERALIZED EQUATIONS LEADING TO SOLUTIONS IN TERMS OF BESSEL FUNCTIONS
BESSEL COEFFICIENTS
INTEGRAL REPRESENTATION OF BESSEL FUNCTIONS
ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR SMALL ARGUMENTS
ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR LARGE ARGUMENTS
INTEGRALS OF BESSEL FUNCTIONS
ZEROES OF BESSEL FUNCTIONS
LEGENDRE FUNCTIONS
LEGENDRE COEFFICIENTS
RECURRENCE FORMULAE FOR LEGENDRE POLYNOMIALS
INTEGRAL REPRESENTATION FOR LEGENDRE POLYNOMIALS
INTEGRALS OF LEGENDRE POLYNOMIALS
EXPANSIONS OF FUNCTIONS IN TERMS OF LEGENDRE POLYNOMIALS
LEGENDRE FUNCTION OF THE SECOND KIND QN(X)
ASSOCIATED LEGENDRE FUNCTIONS
GENERATING FUNCTION FOR ASSOCIATED LEGENDRE FUNCTIONS
RECURRENCE FORMULAE FOR Pnm
INTEGRALS OF ASSOCIATED LEGENDRE FUNCTIONS
ASSOCIATED LEGENDRE FUNCTION OF THE SECOND KIND Qnm

Boundary Value Problems and Eigenvalue Problems
INTRODUCTION
VIBRATION, WAVE PROPAGATION OR WHIRLING OF STRETCHED STRINGS
LONGITUDINAL VIBRATION AND WAVE PROPAGATION IN ELASTIC BARS
VIBRATION, WAVE PROPAGATION AND WHIRLING OF BEAMS
WAVES IN ACOUSTIC HORNS
STABILITY OF COMPRESSED COLUMNS
IDEAL TRANSMISSION LINES (TELEGRAPH EQUATION)
TORSIONAL VIBRATION OF CIRCULAR BARS
ORTHOGONALITY AND ORTHOGONAL SETS OF FUNCTIONS
GENERALIZED FOURIER SERIES
BOUNDARY VALUE PROBLEMS
EIGENVALUE PROBLEMS
PROPERTIES OF EIGENFUNCTIONS OF SELF-ADJOINT SYSTEMS
STURM-LIOUVILLE SYSTEM
STURM-LIOUVILLE SYSTEM FOR FOURTH-ORDER EQUATIONS
SOLUTION OF NON-HOMOGENEOUS EIGENVALUE PROBLEMS
FOURIER SINE SERIES
FOURIER COSINE SERIES
COMPLETE FOURIER SERIES
FOURIER-BESSEL SERIES
FOURIER–LEGENDRE SERIES

Functions of a Complex Variable
COMPLEX NUMBERS
ANALYTIC FUNCTIONS
ELEMENTARY FUNCTIONS
INTEGRATION IN THE COMPLEX PLANE
CAUCHY’S INTEGRAL THEOREM
CAUCHY’S INTEGRAL FORMULA
INFINITE SERIES
TAYLOR’S EXPANSION THEOREM
LAURENT’S SERIES
CLASSIFICATION OF SINGULARITIES
RESIDUES AND RESIDUE THEOREM
INTEGRALS OF PERIODIC FUNCTIONS
IMPROPER REAL INTEGRALS
IMPROPER REAL INTEGRAL INVOLVING CIRCULAR FUNCTIONS
IMPROPER REAL INTEGRALS OF FUNCTIONS HAVING SINGULARITIES ON THE REAL AXIS
THEOREMS ON LIMITING CONTOURS
INTEGRALS OF EVEN FUNCTIONS INVOLVING LOG X
INTEGRALS OF FUNCTIONS INVOLVING Xa
INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS
INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS INVOLVING LOG X
INVERSE LAPLACE TRANSFORMS

Partial Differential Equations of Mathematical Physics
INTRODUCTION
THE DIFFUSION EQUATION
THE VIBRATION EQUATION
THE WAVE EQUATION
HELMHOLTZ EQUATION
POISSON AND LAPLACE EQUATIONS
CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS
UNIQUENESS OF SOLUTIONS
THE LAPLACE EQUATION
THE POISSON EQUATION
THE HELMHOLTZ EQUATION
THE DIFFUSION EQUATION
THE VIBRATION EQUATION
THE WAVE EQUATION

Integral Transforms
FOURIER INTEGRAL THEOREM
FOURIER COSINE TRANSFORM
FOURIER SINE TRANSFORM
COMPLEX FOURIER TRANSFORM
MULTIPLE FOURIER TRANSFORM
HANKEL TRANSFORM OF ORDER ZERO
HANKEL TRANSFORM OF ORDER ν
GENERAL REMARKS ABOUT TRANSFORMS DERIVED FROM THE FOURIER INTEGRAL THEOREM
GENERALIZED FOURIER TRANSFORM
TWO-SIDED LAPLACE TRANSFORM
ONE-SIDED GENERALIZED FOURIER TRANSFORM
LAPLACE TRANSFORM
MELLIN TRANSFORM
OPERATIONAL CALCULUS WITH LAPLACE TRANSFORMS
SOLUTION OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
OPERATIONAL CALCULUS WITH FOURIER COSINE TRANSFORM
OPERATIONAL CALCULUS WITH FOURIER SINE TRANSFORM
OPERATIONAL CALCULUS WITH COMPLEX FOURIER TRANSFORM
OPERATIONAL CALCULUS WITH MULTIPLE FOURIER TRANSFORM
OPERATIONAL CALCULUS WITH HANKEL TRANSFORM

Green’s Functions
INTRODUCTION
GREEN’S FUNCTION FOR ORDINARY DIFFERENTIAL BOUNDARY VALUE PROBLEM
GREEN’S FUNCTION FOR AN ADJOINT SYSTEM
SYMMETRY OF THE GREEN’S FUNCTIONS AND RECIPROCITY
GREEN’S FUNCTION FOR EQUATIONS WITH CONSTANT COEFFICIENTS
GREEN’S FUNCTIONS FOR HIGHER ORDERED SOURCES
GREEN’S FUNCTION FOR EIGENVALUE PROBLEMS
GREEN’S FUNCTION FOR SEMI-INFINITE ONE DIMENSIONAL MEDIA
GREEN’S FUNCTION FOR INFINITE ONE-DIMENSIONAL MEDIA
GREEN’S FUNCTION FOR PARTIAL DIFFERENTIAL EQUATIONS
GREEN’S IDENTITIES FOR THE LAPLACIAN OPERATOR
GREEN’S IDENTITY FOR THE HELMHOLTZ OPERATOR
GREEN’S IDENTITY FOR BI-LAPLACIAN OPERATOR
GREEN’S IDENTITY FOR THE DIFFUSION OPERATOR
GREEN’S IDENTITY FOR THE WAVE OPERATOR
GREEN’S FUNCTION FOR UNBOUNDED MEDIA—FUNDAMENTAL SOLUTION
FUNDAMENTAL SOLUTION FOR THE LAPLACIAN
FUNDAMENTAL SOLUTION FOR THE BI-LAPLACIAN
FUNDAMENTAL SOLUTION FOR THE HELMHOLTZ OPERATOR
FUNDAMENTAL SOLUTION FOR THE OPERATOR, - ∇2 + μ2
CAUSAL FUNDAMENTAL SOLUTION FOR THE DIFFUSION OPERATOR
CAUSAL FUNDAMENTAL SOLUTION FOR THE WAVE OPERATOR
FUNDAMENTAL SOLUTIONS FOR THE BI-LAPLACIAN HELMHOLTZ OPERATOR
GREEN’S FUNCTION FOR THE LAPLACIAN OPERATOR FOR BOUNDED MEDIA
CONSTRUCTION OF THE AUXILIARY FUNCTION-METHOD OF IMAGES
GREEN’S FUNCTION FOR THE LAPLACIAN FOR HALF-SPACE
GREEN’S FUNCTION FOR THE LAPLACIAN BY EIGENFUNCTION EXPANSION FOR BOUNDED MEDIA
GREEN’S FUNCTION FOR A CIRCULAR AREA FOR THE LAPLACIAN
GREEN’S FUNCTION FOR SPHERICAL GEOMETRY FOR THE LAPLACIAN
GREEN’S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR BOUNDED MEDIA
GREEN’S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR HALF-SPACE
GREEN’S FUNCTION FOR A HELMHOLTZ OPERATOR IN QUARTER-SPACE
CAUSAL GREEN’S FUNCTION FOR THE WAVE OPERATOR IN BOUNDED MEDIA
CAUSAL GREEN’S FUNCTION FOR THE DIFFUSION OPERATOR FOR BOUNDED MEDIA
METHOD OF SUMMATION OF SERIES SOLUTIONS IN TWO DIMENSIONAL MEDIA

Asymptotic Methods
INTRODUCTION
METHOD OF INTEGRATION BY PARTS
LAPLACE’S INTEGRAL
STEEPEST DESCENT METHOD
DEBYE’S FIRST ORDER APPROXIMATION
ASYMPTOTIC SERIES APPROXIMATION
METHOD OF STATIONARY PHASE
STEEPEST DESCENT METHOD IN TWO DIMENSIONS
MODIFIED SADDLE POINT METHOD: SUBTRACTION OF A SIMPLE POLE
MODIFIED SADDLE POINT METHOD: SUBTRACTION OF POLE OF ORDER N
SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS FOR LARGE ARGUMENTS
CLASSIFICATION OF POINTS AT INFINITY
SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH REGULAR SINGULAR POINTS
ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK ONE
THE PHASE INTEGRAL AND WKBJ METHOD FOR AN IRREGULAR SINGULAR POINT OF RANK ONE
ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK HIGHER THAN ONE
ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH LARGE PARAMETERS

Numerical Methods
INTRODUCTION
ROOTS OF NON-LINEAR EQUATIONS
ROOTS OF A SYSTEM OF NON-LINEAR EQUATION
FINITE DIFFERENCES
NUMERICAL DIFFERENTIATION
NUMERICAL INTEGRATION
ORDINARY DIFFERENTIAL EQUATIONS: INITIAL VALUE PROBLEMS
ORDINARY DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS
ORDINARY DIFFERENTIAL EQUATIONS: EIGENVALUE PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS

Appendix A: Infinite Series
Appendix B: Special Functions
Appendix C: Orthogonal Coordinate Systems
Appendix D: Dirac Delta Functions
Appendix E: Plots of Special Functions
Appendix F: Vector Analysis
Appendix G: Matrix Algebra

References