Find $\int_{|z+2|=1}\frac{e^{z^2}}{z^{1995}}dz$

Question

Find $\int_{|z+2|=1}\frac{e^{z^2}}{z^{1995}}dz$, where $z\in \mathbb{C}$.

My current struggle with this problem is first just parametrizing $\{z:|z+2|=1\}$.

How can I find a $\gamma:[a,b]\rightarrow \mathbb{C}$ that would give me this curve?

Thanks in advance

Answer 1

The function $f(z) = \frac{e^{z^{2}}}{z^{1995}}$ is analytic in the region $\{z\in\mathbb{C}:|z+2|<1+\epsilon\}$ since its only pole is at $z=0.$ Therefore by Cauchy's theorem,

$$\int_{|z+2|=1}\frac{e^{z^2}}{z^{1995}}dz = 0.$$