Fourier transform of the generalized exponential function

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# Fourier transform of the generalized exponential function

Submitted by DanielWhite on Fri, 10/25/2013 - 04:50

Forums:

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.

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## Versions

(v2) by DanielWhite 2013-10-25

## Clarification

> 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

The generalized exponential function

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

## Further clarification

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

The problem is what happens to $E_{n}(x)$ when $x$ is negative. The integral which defines the function blows up exponentially so is of no use. One can continue $E_{n}$ 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 $x$ 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?