[Math] Computing a contour integral over curve not centered at origin

Question

Consider the integral

$$
\int_C \frac{1}{z} \, dz
$$

where $C$ is the circle of radius $R$ centered at the point $z_0 \in \mathbb{C}$. We parametrize the curve by $z(\theta) = z_0 + Re^{i\theta}$ where $0 \leq \theta \leq 2\pi$. It follows immediately that $dz = iRe^{i\theta} \, d\theta$. Substituting, we obtain

$$
\int_0^{2\pi} \frac{1}{z_0 + Re^{i\theta}} \cdot iRe^{i\theta} \, d\theta = iR \int_0^{2\pi} \frac{e^{i\theta}}{z_0 + Re^{i\theta}} \, d\theta.
$$

Does anyone have any advice on how to proceed in commuting the integral? I suppose this is more of a calculus question than a complex analysis question. I apologize if this is very elementary; my integration is a little rusty.

Answer 1

Hint:

$$\int_{a}^{b}\frac{f'(\theta)}{f(\theta)} \, d\theta = \ln(f(\theta))\Big|_{\theta = a}^{\theta=b} .$$