I need to evaluate $$\int_{|z|=0.6}{\frac{e^z}{ze^z-2iz}dz}$$ using the Cauchy Formula
Let $g(z)=\frac{e^z}{ze^z-2iz}$. $g(z)$ has one pole at $z=0$ which lies inside $|z|=0.6$. But $g(z)$ is not analytic on such circle.
I see that I can write $$\frac{e^z}{ze^z-2iz}=\frac{1}{z}\frac{e^z}{e^z-2i}$$
Let $f(z)=\frac{e^z}{e^z-2i}$ which is analytic over $|z|=0.6$. Thus $$\int_{|z|=0.6}{\frac{e^z}{ze^z-2iz}dz}=2\pi i*f(0)=2\pi i\left(\frac{1}{5}+\frac{2i}{5}\right)$$