Principles of Applied Mathematicsprovides a comprehensive look at how classical methods are used in many fields and contexts. Updated to reflect developments of the last twenty years, it shows how two areas of classical applied mathematicsspectral theory of operators and asymptotic analysisare useful for solving a wide range of applied science problems. Topics such as asymptotic expansions, inverse scattering theory, and perturbation methods are combined in a unified way with classical theory of linear operators. Several new topics, including wavelength analysis, multigrid methods, and homogenization theory, are blended into this mix to amplify this theme.This book is ideal as a survey course for graduate students in applied mathematics and theoretically oriented engineering and science students. This most recent edition, for the first time, now includes extensive corrections collated and collected by the author.

James P. Keener is a professor of mathematics at the University of Utah. He received his Ph.D. from the California Institute of Technology in applied mathematics in 1972. In addition to teaching and research in applied mathematics, Professor Keener served as editor in chief of the SIAM Journal on Applied Mathematics and continues to serve as editor of several leading research journals. He is the recipient of numerous research grants. His research interests are in mathematical biology with an emphasis on physiology. His most recent book, co-authored with James Sheyd, is Mathematical Physiology.

Preface to First Edition	p. xi
Preface to Second Edition	p. xvii
Finite Dimensional Vector Spaces	p. 1
Linear Vector Spaces	p. 1
Spectral Theory for Matrices	p. 9
Geometrical Significance of Eigenvalues	p. 17
Fredholm Alternative Theorem	p. 24
Least Squares Solutions-Pseudo Inverses	p. 25
The Problem of Procrustes	p. 41
Applications of Eigenvalues and Eigenfunctions	p. 42
Exponentiation of Matrices	p. 42
The Power Method and Positive Matrices	p. 43
Iteration Methods	p. 45
Further Reading	p. 47
Problems for Chapter 1	p. 49
Function Spaces	p. 59
Complete Vector Spaces	p. 59
Sobolev Spaces	p. 65
Approximation in Hilbert Spaces	p. 67
Fourier Series and Completeness	p. 67
Orthogonal Polynomials	p. 69
Trigonometric Series	p. 73
Discrete Fourier Transforms	p. 76
Sinc Functions	p. 78
Wavelets	p. 79
Finite Elements	p. 88
Further Reading	p. 92
Problems for Chapter 2	p. 93
Integral Equations	p. 101
Introduction	p. 101
Bounded Linear Operators in Hilbert Space	p. 105
Compact Operators	p. 111
Spectral Theory for Compact Operators	p. 114
Resolvent and Pseudo-Resolvent Kernels	p. 118
Approximate Solutions	p. 121
Singular Integral Equations	p. 125
Further Reading	p. 127
Problems for Chapter 3	p. 128
Differential Operators	p. 133
Distributions and the Delta Function	p. 133
Green's Functions	p. 144
Differential Operators	p. 151
Domain of an Operator	p. 151
Adjoint of an Operator	p. 152
Inhomogeneous Boundary Data	p. 154
The Fredholm Alternative	p. 155
Least Squares Solutions	p. 157
Eigenfunction Expansions	p. 161
Trigonometric Functions	p. 164
Orthogonal Polynomials	p. 167
Special Functions	p. 169
Discretized Operators	p. 169
Further Reading	p. 171
Problems for Chapter 4	p. 171
Calculus of Variations	p. 177
The Euler-Lagrange Equations	p. 177
Constrained Problems	p. 180
Several Unknown Functions	p. 181
Higher Order Derivatives	p. 184
Variable Endpoints	p. 184
Several Independent Variables	p. 185
Hamilton's Principle	p. 186
The Swinging Pendulum	p. 188
The Vibrating String	p. 189
The Vibrating Rod	p. 189
Nonlinear Deformations of a Thin Beam	p. 193
A Vibrating Membrane	p. 194
Approximate Methods	p. 195
Eigenvalue Problems	p. 198
Optimal Design of Structures	p. 201
Further Reading	p. 202
Problems for Chapter 5	p. 203
Complex Variable Theory	p. 209
Complex Valued Functions	p. 209
The Calculus of Complex Functions	p. 214
Differentiation-Analytic Functions	p. 214
Integration	p. 217
Cauchy Integral Formula	p. 220
Taylor and Laurent Series	p. 224
Fluid Flow and Conformal Mappings	p. 228
Laplace's Equation	p. 228
Conformal Mappings	p. 236
Free Boundary Problems	p. 243
Contour Integration	p. 248
Special Functions	p. 259
The Gamma Function	p. 259
Bessel Functions	p. 262
Legendre Functions	p. 268
Sinc Functions	p. 270
Further Reading	p. 273
Problems for Chapter 6	p. 274
Transform and Spectral Theory	p. 283
Spectrum of an Operator	p. 283
Fourier Transforms	p. 284
Transform Pairs	p. 284
Completeness of Hermite and Laguerre Polynomials	p. 297
Sinc Functions	p. 299
Windowed Fourier Transforms	p. 300
Wavelets	p. 301
Related Integral Transforms	p. 307
Laplace Transform	p. 307
Mellin Transform	p. 308
Hankel Transform	p. 309
Z Transforms	p. 310
Scattering Theory	p. 312
Scattering Examples	p. 318
Spectral Representations	p. 325
Further Reading	p. 327
Problems for Chapter 7	p. 328
Fourier Transform Pairs	p. 335
Partial Differential Equations	p. 337
Poisson's Equation	p. 339
Fundamental Solutions	p. 339
The Method of Images	p. 343
Transform Methods	p. 344
Hilbert Transforms	p. 355
Boundary Integral Equations	p. 357
Eigenfunctions	p. 359
The Wave Equation	p. 365
Derivations	p. 365
Fundamental Solutions	p. 368
Vibrations	p. 373
Diffraction Patterns	p. 376
The Heat Equation	p. 380
Derivations	p. 380
Fundamental Solutions	p. 383
Transform Methods	p. 385
Differential-Difference Equations	p. 390
Transform Methods	p. 392
Numerical Methods	p. 395
Further Reading	p. 400
Problems for Chapter 8	p. 402
Inverse Scattering Transform	p. 411
Inverse Scattering	p. 411
Isospectral Flows	p. 417
Korteweg-deVries Equation	p. 421
The Toda Lattice	p. 426
Further Reading	p. 432
Problems for Chapter 9	p. 433
Asymptotic Expansions	p. 437
Definitions and Properties	p. 437
Integration by Parts	p. 440
Laplace's Method	p. 442
Method of Steepest Descents	p. 449
Method of Stationary Phase	p. 456
Further Reading	p. 463
Problems for Chapter 10	p. 463
Regular Perturbation Theory	p. 469
The Implicit Function Theorem	p. 469
Perturbation of Eigenvalues	p. 475
Nonlinear Eigenvalue Problems	p. 478
Lyapunov-Schmidt Method	p. 482
Oscillations and Periodic Solutions	p. 482
Advance of the Perihelion of Mercury	p. 483
Van der Pol Oscillator	p. 485
Knotted Vortex Filaments	p. 488
The Melnikov Function	p. 493
Hopf Bifurcations	p. 494
Further Reading	p. 496
Problems for Chapter 11	p. 498
Singular Perturbation Theory	p. 505
Initial Value Problems I	p. 505
Van der Pol Equation	p. 508
Adiabatic Invariance	p. 510
Averaging	p. 511
Homogenization Theory	p. 514
Initial Value Problems II	p. 520
Operational Amplifiers	p. 521
Enzyme Kinetics	p. 523
Slow Selection in Population Genetics	p. 526
Boundary Value Problems	p. 528
Matched Asymptotic Expansions	p. 528
Flame Fronts	p. 539
Relaxation Dynamics	p. 542
Exponentially Slow Motion	p. 548
Further Reading	p. 551
Problems for Chapter 12	p. 552
Bibliography	p. 559
Selected Hints and Solutions	p. 567
Index	p. 596
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