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Fourier transform of the generalized exponential function

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Fourier transform of the generalized exponential function

Is this fourier transform defined? I study applied physics and hence do more applied mathematics.

This is a little bit beyond my mathematical background, so I thought I’d ask the mathematicians.

Thanks in advanced, and let me know if you need any clarifications of the problem.


> Thanks in advance, and let me know if you need any clarifications of the problem.

What do you mean by "generalized exponential function"?

The generalized exponential functionPlanetmathPlanetmath: E_n (x)

Inegral from 1 to infinity of, exp(-t*x)*(t^-n)*dt

As it is usually defined, the Fourier transform involves an integral over both positive and negative values, i.e.

-+En(x)sin(x)dxsuperscriptsubscriptsubscriptEnxxdx\int_{{-\infty}}^{{+\infty}}E_{n}(x)\sin(x)\,dx

The problem is what happens to En(x)subscriptEnxE_{n}(x) when xxx is negative. The integral which defines the function blows up exponentially so is of no use. One can continue EnsubscriptEnE_{n} analytically from the positive to the negative real axis, but the result is multiply valued and those values are complex.

Thus, before proceeding further, I would like to take a step back and ask about the applied context in which this arises. Is xxx allowed to be negative and, if so, does it make sense to have complex values? Do you need a Fourier transform over the whole real axis or some variant that only goes over the positive values?

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