How to parametrize a circle not centered at the origin

Question

I'm trying to parametrize the circle centered at $2i$ with radius $1$. I'm trying to parametrize it to find the integral of $\frac{1}{z}$ from that circle so I'm not sure if I can put this to make it $e^{iz}$ or how I can do that.

Answer 1

Define$$\begin{array}{rccc}\gamma\colon&[0,2\pi]&\longrightarrow&\Bbb C\\&\theta&\mapsto&2+e^{i\theta}.\end{array}$$Not that it matters a lot, since that integral is clearly equal to $0$.