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absolutely convergent analytic function applying the Laplace arbitrary arccos asymptotic belongs Bessel function Bessel transformable boundary conditions bounded variation complex number constant coefficients convolution theorem cosec cosh defined definition denotes derivative differential equation elements erfc evaluation exists expansion field find the solution finite interval follows Fourier integral Fourier series Fourier transforms func function f(t given half-plane Hankel Hankel transform Hence initial conditions integral 2.1 integral equation integral transforms inversion formula isomorphism Kontorovich-Lebedev transform Laguerre Laplace integral Laplace transform Let f(t linear logarithm Math Meijer Mellin transforms multiplication notation obtain odd function oo oo oo operational calculus operator function polynomial properties reduces ring satisfying the condition sequence solution u(x solving step function theory tion variable whence whilst zero divisors