Residue of function with different branch points I'm wondering what would one do when one wishes to find the residue of a function
$$\text{Res}_{z\to z_0} f(z)$$
where $f(z)$ has multiple branch points, for instance $f(z)$ may be a function such as 
$$f(z) = \frac{1}{\sqrt{z}+\ln z}\quad \text{or} \quad f(z)=\frac{1}{\ln(\sqrt{z}+1)}$$
I'd clearly have to do a Laurent series expansion, but I'm wondering will the branch point(s) affect the validity of my expansion? 
 A: You have to do the expansion using the particular branch you're interested in.  It's quite possible that a point may be a singularity for one branch but not for another, e.g. $f(z) = \dfrac{1}{\sqrt{z} - 1}$ has a pole at $z=1$ for a branch where $\sqrt{1} = 1$, but not for a branch where $\sqrt{1}=-1$.
I hope you're not trying to find a residue at a branch point: that doesn't exist.  You only have a residue when you have an isolated singularity, and a branch point is not an isolated singularity.  Thus your second example has no 
isolated singularities: to make the denominator $0$ you'd need $z=0$, but that's a branch point of the square root.
Your first example $1/(\sqrt{z} + \ln(z))$ will have poles where $\sqrt{z} + \ln(z) = 0$, namely  $4 W(\pm 1/2)^2$ where $W$ is a branch of the Lambert W function.  One such point (using the "main" branch $W_0$ and $+1/2$) is approximately $0.4948664144$, and requires using the principal branches of $\sqrt{z}$ and $\ln(z)$.  If that point is $p_0$, we have 
$$\sqrt{z} + \ln(z) = \left(\dfrac{1}{2 \sqrt{p_0}} + \dfrac{1}{p_0}\right) (z - p_0) + O\left((z-p_0)^2\right)$$
so the residue at $p_0$ is $$\dfrac{1}{\dfrac{1}{2 \sqrt{p_0}} + \dfrac{1}{p_0}}$$
using the same branch of $\sqrt{p_0}$ that you are using for $\sqrt{p_0} + \ln(p_0) = 0$.
