Approximate Solution of Linear Differential Equations

  • Carl M. Bender
  • Steven A. Orszag


The theory of linear differential equations is so powerful that one can usually predict the local behavior of the solutions near a point x 0 without knowing how to solve the differential equation. It suffices to examine the coefficient functions of the differential equation in the neighborhood of x 0.


Singular Point Linear Differential Equation Leading Behavior Asymptotic Series Asymptotic Relation 
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Chapter 3 For a general discussion see

  1. 15.
    Erdelyi, A., Asymptotic Expansions, Dover Publications, Inc., New York, 1956.zbMATHGoogle Scholar
  2. 16.
    de Bruijn, N. G., Asymptotic Methods in Analysis, North-Holland Publishing Company, Amsterdam, 1958.zbMATHGoogle Scholar
  3. 17.
    Carrier, G. F., Krook, M, and Pearson, C. E., Functions of a Complex Variable, McGraw-Hill Book Company, Inc., New York, 1966.zbMATHGoogle Scholar
  4. 18.
    Heading, J., An Introduction to Phase-Integral Methods, Methuen and Co., Ltd., London, 1962.zbMATHGoogle Scholar
  5. 19.
    Wasow, W. A., Asymptotic Expansions for Ordinary Differential Equations, John Wiley and Sons, Inc., New York, 1965. Also see Refs. 4 and 5.zbMATHGoogle Scholar

Chapter 3 For asymptotic expansions of the special functions of mathematical physics and applied mathematics see Refs. 10 to 12 and

  1. 20.
    Watson, G. N., A Treatise on the Theory ofBessel Functions, 2d ed., Cambridge University Press, Cambridge, 1944.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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