Multi-value integration

Question

Is there any place which demonstrate how an integration over a Riemann surface (a branch-cut for example) can lead for a result which involves multiple different values? I looked in several places but could not find any sufficiently clear source.

Answer 1

I think the best example to contemplate is $$ \log z = \int_1^z\frac{dz}{z} $$ Now if $z$ is a (nonzero) complex number, the result here depends on the path you choose going from $1$ to $z$. The different values are the branches of $\log z$.

After understanding that, we can consider branches of $z^r$ where $r$ is rational. And perhaps then do branches of arcsin, where the Riemann surface is more complicated, with branch points at $1$ and $-1$ ... $$ \arcsin z = \int_0^z\frac{dx}{\sqrt{1-z^2}} $$ Now the path of integration is not in $\mathbb C$ but in the Riemann surface of $(1-z^2)^{-1/2}$.