TITLE: Calculate $ \int_\gamma \frac{1}{z-i}dz $
QUESTION [5 upvotes]: Calculate the following integral
$$
\int_\gamma \frac{1}{z-i}dz
$$
with $\gamma$ circle of radius 2 centered at the origin. Any suggestions please? Thank you very much.

REPLY [5 votes]: The question is pretty basic so I assume you're beginning with this stuff and you probably want to go with basic line integrals, parametrization and stuff. 
Parametrize thus...but with a twist 
$$z-i=2e^{it}\;,\;\;0\le t\le 2\pi\implies dz=2ie^{it}\,dt\implies$$
$$\oint\limits_C\frac{dz}{z-i}=\int\limits_0^{2\pi}\frac{2ie^{it}dt}{2e^{it}}=\int\limits_0^{2\pi}i\,dt=2\pi i$$

REPLY [2 votes]: By the Residue theorem we have
$$
\int_\gamma \frac{1}{z-i}dz=2i\pi Res(f,i)=2i\pi
$$