The Mehler-Fock transform of general order and arbitrary index and its inversion

Cyril Nasim

International Journal of Mathematics and Mathematical Sciences, 1984, vol. 7, 1-10

Abstract:

An integral transform involving the associated Legendre function of zero order, P − 1 2 + i τ ( x ) , x ∈ [ 1 , ∞ ) , as the kernel (considered as a function of τ ), is called Mehler-Fock transform. Some generalizations, involving the function P − 1 2 + i τ μ ( x ) , where the order μ is an arbitrary complex number, including the case when μ = 0 , 1 , 2 , … have been known for some time. In this present note, we define a general Mehler-Fock transform involving, as the kernel, the Legendre function P − 1 2 + t μ ( x ) , of general order μ and an arbitrary index − 1 2 + t , t = σ + i τ , − ∞ < τ < ∞ . Then we develop a symmetric inversion formulae for these transforms. Many well-known results are derived as special cases of this general form. These transforms are widely used for solving many axisymmetric potential problems.

Date: 1984
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:650256

DOI: 10.1155/S016117128400017X

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