Question

Q: Evaluate the following integral using residue theorem

∫z^2 e^z dz ; C : |z| = 1

Answer 1

Consider the integral $\int_C z^2e^z$ where $C=\{z\in\mathbb C: |z|=1\}$. The function $f(z)=z^2e^z$ has no a singular point, and is a holomorphic function in $D=\{z\in\mathbb C: |z|\leq 1\}$. Therefore, Cauchy’s residue theorem implies $\int_C z^2e^z =0.$