Inverse Transforms of Products of Legendre Transforms
نویسندگان
چکیده
is obtained by applying successive integration by parts to the integral represented by the left-hand member of this equation so that the new integrand is the product of F by [(1 —x2)P„' ]', and by replacing this second factor by — n(n + l)Pn according to Legendre's differential equation [l]. Thus certain boundary value problems in ordinary and partial differential equations that involve the differential form appearing in the braces of equation (2) reduce to problems containing one less independent variable when written in terms of the transform of the unknown function. This has been illustrated in the literature [l; 2]. The role played by boundary conditions and differential forms in determining appropriate integral transforms for the reduction of given types of boundary value problems was indicated in an earlier paper [3]. The inverse F_1 {/(«)} denotes a function F(x) whose transform is /(«). If F(x) satisfies conditions under which it is represented by its Legendre series [4], then it follows from the formulas for the coefficients in that series that
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