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I have a family of functions defined by integrals and indexed by n, e.g

f[x_,n_]=[itex]\int dy e^{ixy}y^n[/itex]

Is it possible to evaluate the integrals corresponding to different particular values of n in such a way that mathematica "remembers" that say f[x,4]= some function g[x]?

An additional complication is that my integrands contain Heaviside theta functions of non-trivial arguments, e.g. [itex]\theta(1-y^2)[/itex] that mathematica doesn't seem to like, and so far I've only been able to proceed by breaking up the integration region manually; I'm hoping to work out a way of reparametrising the expression so that I don't have to do this, but if I can't then I'd need a way of piecing different functions together and getting mathematica to understand that the result is equivalent to the original integral.

Thanks in advance.