Article
A modified finite Hankel transform
Integral Transforms and Special Functions (Impact Factor: 0.72). 10/2003; 14(5):403412. DOI: 10.1080/10652460310001600654
ABSTRACT
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, Parsevaltype 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.

 "It is given in the form H t ½f ðtÞ; k ¼ Z b a tf ðtÞJ t ðtkÞdt; where f(t) is a function defined in a finite interval ½a; b, satisfying the Dirichlet's conditions and J t is the Bessel function of first kind and of order t. Furthermore, Khajah [12] has studied a modified form of Hankel transform under two assumptions on the parameter. In each case the inversion formula, Parsevaltype identity, transform of derivatives and transforms of products of the kind t k f ðtÞ has been derived. "
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