# complex analysis – What type of singularity is \$-i\$ for \$f(z)=frac{sqrt{z}}{z+i}\$, where \$sqrt{z}\$ is the principal branch of \$sqrt{z}\$?

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 $$0<|z+i|0$$? 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)