How do you derive an explicit solution to a functional equation using complex analysis?

Question

For instance, take the gamma function. Somehow there is a way to use the identity $f(z+1)= z f(z)$ and the condition $f(1)=1$ to derive the Euler/Laplace integral representation of the gamma function $\Gamma(z)$ using complex analysis as a solution to the functional equation, but how is that done? From what I have seen it is some kind of complicated analytical process involving contour integrals, can you apply a similar process to other functional identities?