Evaluate the contour integration

Question

Evaluate the below integral:

$$\oint_{c}\frac{e^{2z}}{(z+1)^4}dz$$

where $C$ is the circle, $|z|=3$

Answer 1

From Cauchy's integral formula,

$$\frac{2\pi i}{n!} f^{n}(z_{0})=\oint_{c}\frac{f(z)}{(z-z_{0})^{n+1}}$$

so from your question, $f(z)=e^{2z}, z_{0}=-1, n=3$

hence, $\frac{2\pi i}{3!} f^{(3)} (-1) = \frac{8\pi i}{3e^{2}}$

Answer 2

The function $e^{2z}$ is analtyic inside $|z| = 3|$, so

$$\oint_{c}\frac{e^{2z}}{(z+1)^4}dz = \frac{2\pi i f^{(3)}(-1)}{3!} = \frac{2^4\pi ie^{-2}}{3!} = \frac{8\pi ie^{-2}}{3}.$$