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,
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):
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.