Evaluate $\int_{C}\frac{|z|\mathit{e}^{z}}{z^{2}}dz$.

Question

Evaluate $\int_{C}\frac{|z|\mathit{e}^{z}}{z^{2}}dz$, where $C$ is the circumference of the circle of radius 2 around the origin.

I wanted to use the Cauchy integral formula, but is $|z|$ analytic on the circle? The answer is $4\pi i$.

Any advice on which theorem I must use? Thanks!

Answer 1

The function $|z|$ is not analytic on the circle, but it is constant ($z=2$) on the circle centered at the origin and of radius $2$!

So

$$\int_{C}\frac{|z|\mathit{e}^{z}}{z^{2}}\,dz=2\int_{C}\frac{\mathit{e}^{z}}{z^{2}}\,dz=2\cdot 2\pi i \operatorname{Res}_{z=0} \frac{\mathit{e}^{z}}{z^{2}}=4\pi i.$$