Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior.

Question

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior.

my works

I'm stuck in next step

Answer 1

You need the coefficient of $\frac{1}{z^3}$ from the series multiplication. There are two ways you can get $\frac{1}{z^3}$. $n=3$ and $m=0$ or $n=1$ and $m=1$ Sum the product of those coefficients and you'll get $$\frac 16 \frac11 + \frac 11 \frac 12=\frac23$$

So the result is

$$\frac 43 \pi i$$

Answer 2

Hint: Do not expand $\cosh{(1/z)}$ as a series: just use the definition: $\cosh{(1/z)}=\frac{e^{1/z}+e^{-1/z}}{2}$. That way you will get rid of the double summation.