Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation TheoryThis book gives a clear, practical and selfcontained presentation of the methods of asymptotics and perturbation theory and explains how to use these methods to obtain approximate analytical solutions to differential and difference equations. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form and for which bruteforce numerical methods may not converge to useful solutions. The objective of this book is to teaching the insights and problemsolving skills that are most useful in solving mathematical problems arising in the course of modern research. Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with a an exposition of global asymptotic methods, including boundarylayer theory, WKB theory, and multiplescale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many wellchosen examples to teach the reader how an applied mathematician tackles problems. There are 190 computergenerated plots and tables comparing approximate and exact solutions; over 600 problems, of varying levels of difficulty; and an appendix summarizing the properties of special functions. 
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Review: Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory
User Review  Joecolelife  GoodreadsI have learned a lot of asymptotic methods from this book, having spent several previous years wondering what that stationary phase method that the physicsts were on about was (by the way B&O poopoo ... Read full review
Contents
II  3 
III  5 
IV  7 
V  11 
VI  14 
VII  20 
VIII  24 
IX  27 
XLV  276 
XLVI  280 
XLVII  302 
XLVIII  306 
XLIX  319 
L  324 
LI  330 
LII  335 
X  29 
XI  30 
XII  36 
XIII  37 
XIV  40 
XV  49 
XVI  53 
XVIII  61 
XIX  62 
XX  66 
XXI  68 
XXII  76 
XXIII  88 
XXIV  103 
XXV  107 
XXVI  118 
XXVII  136 
XXVIII  146 
XXIX  148 
XXX  152 
XXXI  171 
XXXII  185 
XXXIII  196 
XXXIV  205 
XXXV  206 
XXXVI  214 
XXXVII  218 
XXXVIII  227 
XXXIX  233 
XL  240 
XLI  247 
XLII  249 
XLIII  252 
XLIV  261 
Common terms and phrases
Airy function analysis analytic approximation to y(x asymptotic analysis asymptotic expansion asymptotic matching asymptotic relation asymptotic series behavior of solutions behavior of y(x boundary conditions boundary layer boundarylayer theory boundaryvalue problem branch points Clue coefficients compute constant contour controlling factor critical point derive determine difference equation differential equation eigenvalue problem exact solution Example exponentially Find the leading finite firstorder formula Frobenius series gives higherorder initial conditions initialvalue problem inner integral representation irregular singular point Laplace's method leading behavior leadingorder linearly independent linearly independent solutions local analysis nonlinear obtain optimal asymptotic approximation outer solution Pade approximants Pade sequence parameter perturbation problem perturbation series perturbation theory plot poles polynomial Prob radius of convergence result saddle point satisfies secondorder Show solution y(x solve steepest steepestdescent Stirling series Substituting Taylor series tion trajectories uniform approximation valid values vanishes verify WKB approximation WKB theory