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A modified finite Hankel transform

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A modified Hankel transform in the form $${\sf J}_\mu \lsqb\, f\lpar z\rpar \semicolon \; s\comma \; \lambda\rsqb \equiv \vint_0^b z^\lambda f\lpar z\rpar \,J_\mu \lpar z s\rpar \, \hbox{d}z$$ is introduced, where f(z) satisfies Dirichlet's conditions in the interval [0, b]. This transform is treated under two assumptions on the parameter s: (i) where s is a root of the transcendental equation J(bu) = 0, and (ii) where s is a root of the transcendental equation u J(bu) + h J(bu) = 0 for a positive constant h. In each case, we derive the inversion formulas, Parseval-type identities, transforms of derivatives, as well as transforms of products of the form z f(z). Some special cases are given together with the transform of a differential operator. Our results are consistent with those established for  = 1.

Keywords: Hankel transform; Integral transform; finite Hankel transform

Document Type: Research Article


Publication date: October 1, 2003

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