Abstract and Applied Analysis

The Analysis of Contour Integrals

Tanfer Tanriverdi and JohnBryce Mcleod

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For any n , the contour integral y = cosh n + 1 x C ( cosh ( z s ) / ( sinh z - sinh x ) n + 1 d z, s 2 = - λ , is associated with differential equation d 2 y ( x ) / d x 2 + ( λ + n ( n + 1 ) / cosh 2 x ) y ( x ) = 0 . Explicit solutions for n = 1 are obtained. For n = 1 , eigenvalues, eigenfunctions, spectral function, and eigenfunction expansions are explored. This differential equation which does have solution in terms of the trigonometric functions does not seem to have been explored and it is also one of the purposes of this paper to put it on record.

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Abstr. Appl. Anal., Volume 2008 (2008), Article ID 765920, 12 pages.

First available in Project Euclid: 9 September 2008

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Tanriverdi, Tanfer; Mcleod, JohnBryce. The Analysis of Contour Integrals. Abstr. Appl. Anal. 2008 (2008), Article ID 765920, 12 pages. doi:10.1155/2008/765920. https://projecteuclid.org/euclid.aaa/1220969152

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  • E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, Oxford University Press, Oxford, UK, 2nd edition, 1962.
  • E. Kamke, Differentialgleichhungen, Akademische Verlagsgesellschaft, Leipzig, Germany, 1943.
  • S. P. Hastings and J. B. McLeod, “Travelling waves and steady solutions for a discrete reaction-diffusion equation,” Preprint.
  • P. C. Fife and J. B. McLeod, “The approach of solutions of nonlinear diffusion equations to travelling front solutions,” Archive for Rational Mechanics and Analysis, vol. 65, no. 4, pp. 335–361, 1977.
  • P. C. Fife and J. B. McLeod, “A phase plane discussion of convergence to travelling fronts for nonlinear diffusion,” Archive for Rational Mechanics and Analysis, vol. 75, no. 4, pp. 281–314, 1981.
  • A. Carpio, S. J. Chapman, S. P. Hastings, and J. B. McLeod, “Wave solutions for a discrete reaction-diffusion equation,” European Journal of Applied Mathematics, vol. 11, no. 4, pp. 399–412, 2000.
  • T. Tanriverdi, Boundary-value problems in ODE, Ph.D. thesis, University of Pittsburgh, Pittsburgh, Pa, USA, 2001.
  • T. Tanriverdi and J. B. Mcleod, “Generalization of the eigenvalues by contour integrals,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1765–1773, 2007.