What type of singularity is $-i$ for $f(z)=frac{sqrt{z}}{z+i}$, where $sqrt{z}$ is the principal branch of $sqrt{z}$. How to find the Laurent series of $f(z)$ for $0<|z+i|<R, R>0$? Or is there another way to decide the residue of $f$ at $-i$?

My goal is using Cauchy’s residue theorem to calculate $int_{+C}f(z)dz$, where $+C$ is a simple closed circle about $-i$ positively oriented. (I know parametrizing the contour can give the value of the integral)