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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
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Inversion of Moment-generating functions (aka Laplace transform of prob dist)
The inverse Laplace transform takes place in the complex domain and includes the residue theorem. You can probably do that analytically (as the above forwarded you to Wiki). But usually you can perhap …