Consider the gamma function $\Gamma(z)=\int_0^{\infty}x^{z-1}e^{-x}dx$, here $\Re(z)>0$.
Let $C$ be the path from $0$ to $ti$ and then from $ti$ to $\infty+ti$, where $t>0$ is a fixed number.
If I define $\Gamma_1(z)=\int_Cs^{z-1}e^{-s}ds$, here the $\log(s)$ is the continuation of the real $\log(x)$ on $\{x>0\}$ along the counterclockwise direction.
Then would the two function equals, or do they have any relations? Especially when $0<\Re(z)<1$.
They will coincide for ${\rm Re}\,z>0$ since the integral is convergent in this range of $z$.
Actually, we have the full course on complex analysis on edX. https://www.edx.org/course/complex-analysis The registration opened two days ago. It has a special chapter on analytic continuation with the help of contour deformation.
The welcome video on the course is here: https://youtu.be/2aHEqRUYBUE