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Methods Based on Cauchy Integrals

  • B. Davies
Part of the Applied Mathematical Sciences book series (AMS, volume 25)

Abstract

The major difficulty in using the Wiener-Hopf technique is the problem of constructing a suitable factorization. We consider here a method based on contour integration which leads by natural extensions to the use of Cauchy integrals in the solution of mixed boundary-value problems.

Keywords

Holder Condition Cauchy Integral Entire Complex Plane Angular Density Plemelj Formula 
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Footnotes

  1. 1.
    See NOBLE (1958), p. 93ff. for details.Google Scholar
  2. 2.
    C. Mark, Phys. Rev. (1947), 72, 558; G. Placzek, Phys. Rev. (1947), 72, 556.Google Scholar
  3. 3.
    MUSKHELISHVILI (1953).Google Scholar
  4. 4.
    We could in fact use a Fourier inversion contour which is not a straight line parallel to the real axis, thus achieving a generalization of the Wiener-Hopf technique by using the Plemelj formula.Google Scholar
  5. 5.
    MUSKHELISHVILI (1953), Ch. 2.Google Scholar
  6. 6.
    MUSKHELISHVILI (1953), Ch. 4.Google Scholar
  7. 7.
    We could in fact use a Fourier inversion contour which is not a straight line parallel to the real axis, thus achieving a generalization of the Wiener-Hopf technique by using the Plemelj formula.Google Scholar
  8. 8.
    Based on work by K. M. Case and R. D. Hazeltine, J. Math. Phys. (1971), 12, 1970.Google Scholar
  9. 9.
    See CASE $ ZWEIFEL (1967) for the derivation and inter- pretation of this equation.Google Scholar
  10. 11.
    Proved in CASE & ZWEIFEL (1967), p. 62 ff.Google Scholar
  11. 12.
    See ref. 8 for another example.Google Scholar
  12. 13.
    Based on a paper by K. M. Case, Rev. Mod. Phys. (1964), 36, 669.Google Scholar
  13. 14.
    See Section 10.4.Google Scholar
  14. 15.
    The techniques used for the solution of these singular integral equations are quite standard; see MUSKHELISHVILI (1953).Google Scholar
  15. 16.
    P. Wolfe, SIAM J. Appl. Math. (1972), 23, 118.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.The Australian National UniversityCanberraAustralia

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