# 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.
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