$\int_{|z|=2}\frac{1+|e^z|\operatorname{Im}z}{z}dz$

Question

Calculate $\int_{|z|=2}\frac{1+|e^z|\operatorname{Im}z}{z}dz$ I know that the curve is $2e^{it}$ and tried to use Cauchy theorem but $0$ is problematic, and I can't. Then I tried breaking up the integral like that $\int_{|z|=2}\frac{1}{z}dz +\int_{|z|=2} \frac{|e^z|\operatorname{Im}z}{z}dz$, but I can't solve the second one.

Any hints ?

Answer 1

You cannot apply Cauchy's theorem, since that's not an analytic function.

On the other hand\begin{align}\require{cancel}\int_{|z|=2}\frac{1+|e^z|\operatorname{Im}z}z\,\mathrm dz&=\int_0^{2\pi}\frac{1+\left|e^{2e^{i\theta}}\right|2\sin\theta}{\cancel{2e^{i\theta}}}\cancel{2e^{i\theta}}i\,\mathrm d\theta\\&=i\int_0^{2\pi}1+2e^{2\cos\theta}\sin(\theta)\,\mathrm d\theta\\&=2\pi i.\end{align}