Asymptotic Expansion of Integrals

  • Carl M. Bender
  • Steven A. Orszag


The analysis of differential and difference equations in Chaps. 3 to 5 is pure local analysis; there we predict the behavior of solutions near one point, but we do not incorporate initial-value or boundary-value data at other points. As a result, our predictions of the local behavior usually contain unknown constants. However, when the differential or difference equation is soluble, we can use the boundary and initial data to make parameter-free predictions of local behavior.


Saddle Point Asymptotic Expansion Integral Representation Steep Descent Local Analysis 
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  1. 25.
    Some general texts on the asymptotic expansion of integrals: Bleistein, N., and Handelsman, R. A., Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, 1975.zbMATHGoogle Scholar
  2. 26.
    Copson, E. T., Asymptotic Expansions, Cambridge University Press, Cambridge, 1967.Google Scholar
  3. 27.
    Olver, F. W. J., Asymptotics and Special Functions, Academic Press, Inc., New York, 1974.Google Scholar
  4. 28.
    See also Refs. 15 to 17. Integral representations of special functions are given in Refs. 10 to 12 and: Gradshteyn, I. S., and Ryzhik, I. W., Tables of Integrals, Series, and Products, 4th ed., Academic Press, Inc., New York, 1965.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Carl M. Bender
    • 1
  • Steven A. Orszag
    • 2
  1. 1.Department of PhysicsWashington UniversitySt. LouisUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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