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Hi I have an exercise in a course I am currently taking and a portion of the question involves evaluating an integral of the following form,

$\int{_0^{i\infty}} \exp(-x)x^{t-1}$

which looks identical to the definition of the gamma function with the exception of the bounds.

I have tried modifying the integral to (which is identical on the imaginary axis):

$\int{_0^{i\infty}} \exp(-x)x^{t-1}\Theta(x)(\frac{1}{2}+\frac{i}{2})^{-1}$

and extending contours from $(a,0)$ to $(\infty,0)$ to $(0,i\infty)$ to $(-\infty,0)$ to $(-a,0)$ to $(0,a)$ and back in an effort to get a gamma function from the first path. However, the upper contour is non convergent in the radius of the contour, so this does not seem to be a useful procedure. It would be awesome if someone could point me towards a useful resource for evaluating this integral.

Thanks for your time.

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Maybe you can start with the initial integral and use a contour from (a,0) to ($\infty$, 0) to (0,i$\infty$) to (0,a) to (a,0).

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