The Fourier inversion theorem holds for all Schwartz functions ( roughly speaking, smooth functions that decay quickly and whose derivatives all decay quickly ).
12.
In mathematics such heuristic arguments are not permitted, and the Fourier inversion theorem includes an explicit specification of what class of functions is being allowed.
13.
However, there is no " best " class of functions to consider so several variants of the Fourier inversion theorem exist, albeit with compatible conclusions.
14.
If the function is absolutely integrable in one dimension ( i . e . ) and is piecewise smooth then a version of the Fourier inversion theorem holds.
15.
If the function is absolutely integrable in one dimension ( i . e . ) but merely piecewise continuous then a version of the Fourier inversion theorem still holds.
16.
If is continuous and absolutely integrable on then the Fourier inversion theorem still holds so long as we again define the inverse transform with a smooth cut off function i . e.
17.
I suspect that the best you will get ( other than a numerical answer to a certain degree of accuracy ) will be an infinite series expansion by using the Lagrange inversion theorem.
18.
This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform ( see Mellin inversion theorem ).
19.
Since \ mathcal { F } ^ {-1 } is so similar to \ mathcal { F }, this follows very easily from the Fourier inversion theorem ( changing variables ):
20.
The result was re-derived ( with attribution ) a number of times, most notably by I . J . Good who derived it from his multilinear generalization of the Lagrange inversion theorem.
