# The Operational Calculus of Legendre Transforms

@article{Churchill1954TheOC, title={The Operational Calculus of Legendre Transforms}, author={R. V. Churchill}, journal={Journal of Mathematics and Physics}, year={1954}, volume={33}, pages={165-178} }

When the integral T{R[F]} is integrated successively by parts and -n(n + I)Pn(x) is substituted for R[Pn(x)] in accordance with Legendre's differential equation, the following result is easily obtained. THEOREM 1: Let F(x) denote a function that satisfies these conditions: F'(x) is continuous and F"(x) is bounded and integrable over each interval interior to the interval -1 < x < 1; T {F(x)} exists and limx_±l (1 x2)F(x) = lim"_±l (1 x2)F'(X) = 0.

#### 23 Citations

THE CONTINUOUS LEGENDRE TRANSFORM, ITS INVERSE TRANSFORM, AND APPLICATIONS

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- 1989

This paper develops an operational calculus for the continuous Legendre transform introduced and studied by Butzer, Stens and Wehrens (1). It is an extension of the work done by Churchill et al (2),… Expand

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In this paper we introduce two integral transforms involving the Legendre function in the kernel (see the operators I0+α,β,μ,v and I−α,β,μ,v. defined below) which generalize the classical Liouville… Expand

The solution of the Laplace equation with the Robin boundary conditions : Applications to inverse problems

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This paper studies the following problem with azimuthal symmetry: ∆u=0 in a unit sphere ∂u(r, ζ)/∂r|r=1 + h u(1, ζ)= f(ζ) on a unit sphere, ∆ is the Laplace operator. ζ=cos(θ), θ is the azimuthal… Expand

The continuous associated legendre transform

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The purpose of this paper is to define the continuous associated Legendre transform as an extension of the continuous and the discrete Legendre transform and the discrete associated Legendre… Expand

The analytical solution of the Laplace equation with the Robin boundary conditions on a sphere: Applications to some inverse problems

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This paper studies the third boundary problem of the Laplace equation with azimuthal symmetry.Many solutions of the boundary value problems in spherical coordinates are available in the form of… Expand

A coordinate transform method for one-speed neutron transport in composite slabs

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