There's probably simple solution but... I have a contour integral of the form $\int _{-i \infty}^{+i \infty} f(t) \ dt$. I want to make a transformation $t = g(s)$ so that the integral is real and of the form: \begin{eqnarray} \int_{0}^{\infty} f(g(s)) \ g'(s) \ ds \end{eqnarray}
Thought: the contour $[-i \infty , i \infty]$ is the imaginary axis including the point at infinity, it is geometrically a circle. I'm struggling to find a transformation so that this circle becomes something which is geometrically just a line $[0, \infty]$.
Thank you
Does the formula $t=i\log s$ work? or $t=i(s-1/s)$?