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A systematic formula for the asymptotic expansion of singular integrals. (English) Zbl 0626.45018

The author presents a new method to obtain the asymptotic expansion of an integral with homogeneous kernel defined in terms of its finite part (FP) by considering the integral \(FP\int_{D}f(x)K(x,\epsilon)dx.\) The usefulness of the formula is depicted by two specific examples. In the first example the weight function has an explicit expression, and the other example illustrates the method with respect to a problem of applied nature in which the weight function is unknown.
Reviewer: R.K.Raina

MSC:

45L05 Theoretical approximation of solutions to integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30E15 Asymptotic representations in the complex plane
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[1] C. M. Bender, S. A. Orzag,Advanced Mathematical Methods for Scientists and Engineers. Mac Graw Hill, New York 1978, pp. 335-349.
[2] N. Bleistein, R. A. Handelsman,Asymptotic expansion of integrals. Holt Rinehart and Winston, New York 1975, pp. 102-146. · Zbl 0327.41027
[3] M. P. Brandao,Improper Integrals in Theoretical Aerodynamics-A New Chapter, paper to be presented to the A.I.A.A J., 1987.
[4] A. Erdelyi,Asymptotic Expansions. Dover Publ.; New York 1956, pp. 25-56.
[5] J. Hadamard,Lectures on Cauchy’s Problem in Linear Differential Equations. Dover Publ.; New York 1952, pp. 117-158. · Zbl 0049.34805
[6] T. Kida, Y. Miyai,An alternative treatment of the lifting-line as a perturbation problem. Journal of Applied Mathematics and Physics (ZAMP),29, 591-607, 1978. · Zbl 0399.76013
[7] T. F. Ogilvie,Singular Perturbation Problems in Ship Hydrodynamics. Adv. Appl. Mech., 1977. · Zbl 0471.76048
[8] M. D. van Dyke,Perturbation Methods in Fluid Dynamics. Academic Press, New York 1975, pp. 167-176. · Zbl 0329.76002
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