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Methods Based on Cauchy Integrals

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Integral Transforms and Their Applications

Part of the book series: Applied Mathematical Sciences ((AMS,volume 25))

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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.

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Footnotes

  1. See NOBLE (1958), p. 93ff. for details.

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  2. C. Mark, Phys. Rev. (1947), 72, 558; G. Placzek, Phys. Rev. (1947), 72, 556.

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  3. MUSKHELISHVILI (1953).

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  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.

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  5. MUSKHELISHVILI (1953), Ch. 2.

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  6. MUSKHELISHVILI (1953), Ch. 4.

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  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.

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  8. Based on work by K. M. Case and R. D. Hazeltine, J. Math. Phys. (1971), 12, 1970.

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  9. See CASE $ ZWEIFEL (1967) for the derivation and inter- pretation of this equation.

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  10. Proved in CASE & ZWEIFEL (1967), p. 62 ff.

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  11. See ref. 8 for another example.

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  12. Based on a paper by K. M. Case, Rev. Mod. Phys. (1964), 36, 669.

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  13. See Section 10.4.

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  14. The techniques used for the solution of these singular integral equations are quite standard; see MUSKHELISHVILI (1953).

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  15. P. Wolfe, SIAM J. Appl. Math. (1972), 23, 118.

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Authors and Affiliations

  1. The Australian National University, Post Office Box 4, 2600, Canberra, A.C.T., Australia

    B. Davies

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  1. B. Davies
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© 1978 Springer Science+Business Media New York

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Davies, B. (1978). Methods Based on Cauchy Integrals. In: Integral Transforms and Their Applications. Applied Mathematical Sciences, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-5512-1_19

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  • DOI: https://doi.org/10.1007/978-1-4757-5512-1_19

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-90313-2

  • Online ISBN: 978-1-4757-5512-1

  • eBook Packages: Springer Book Archive

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