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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