From the reviews: "A good introduction to a subject important for its capacity to circumvent theoretical and practical obstacles, and therefore particularly prized in the applications of mathematics. The book presents a balanced view of the methods and their usefulness: integrals on the real line and in the complex plane which arise in different contexts, and solutions of differential equations not expressible as integrals. Murray includes both historical remarks and references to sources or other more complete treatments. More useful as a guide for self-study than as a reference work, it is accessible to any upperclass mathematics undergraduate. Some exercises and a short bibliography included. Even with E.T. Copson's "Asymptotic" "Expansions" or N.G. de Bruijn's "Asymptotic Methods in" "Analysis" (1958), any academic library would do well to have this excellent introduction." ("S. Puckette, University of" "the South") #"Choice Sept. 1984"#1
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LAPLACES METHOD FOR INTEGRALS
METHOD OF STEEPEST DESCENTS
4 other sections not shown
Airy analysis analytic applicable appropriate asymptotic approximation asymptotic expansion asymptotic power series asymptotic sequence asymptotic solutions becomes branch line branch point called choose complex consider constant continuous contour contribution convergent course defined deformed depend determine differential equations discussed domain dominant term example exercise exponential fact finally function given gives hence illustrated immediately important infinity integral interest inversion Laplace's method lies limit linear look maximum method namely neighbourhood Note obtain original oscillations oscillatory passes path positive practical problem procedure radius of convergence range real axis region relation respectively result saddle-point similar simply singularity situation specific steepest descents path substituting Suppose term transform transition usual valid variable vicinity Watson's lemma wave write zero