Vol. 52, No. 2, 1974

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ISSN: 0030-8730
On an inversion theorem for the general Mehler-Fock transform pair

Peter Michael Rosenthal

Vol. 52 (1974), No. 2, 539–545
Abstract

Let Pmk(y) be the Legendre function of the first kind and let Γ(z) be the Gamma function. Then the general Mehler-Fock transform of complex order k of a function g(y) is defined by the equation

              −1           1
f(x) = L2(g) = π x sinh(πx)Γ (2 − k− ix)    ∫
1           ∞      k
× Γ (2 − k+ ix) 1 g(y)P ix−1∕2(y)dy,
the inversion theorem states
             ∫ ∞
g(y) = L1(f ) =   f (x)P k   (y)dx.
0      ix−1∕2

It is stated on page 416 of I. N. Sneddon’s book ‘The Use of Integral Transforms, (1972) that apparently a class of functions g(y) for which this result is valid is not yet clearly defined. The purpose of this paper is to define a class of functions g(y) as well as a class f(x) and give proofs that the above inversion formula hold for these classes.

Mathematical Subject Classification 2000
Primary: 44A15
Milestones
Received: 2 July 1973
Published: 1 June 1974
Authors
Peter Michael Rosenthal