A Modified Finite Hankel Transform
A modified Hankel transform in the form [Formula: See Text] 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 μ(b u) = 0, and (ii) where s is a root of the transcendental equation u J′μ(b u) + h J μ(b u) = 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 δ f(z). Some special cases are given together with the transform of a differential operator. Our results are consistent with those established for λ = 1.
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