For $n \in \mathbb{N}$, evaluate $\displaystyle{\int_{|z|=1} \dfrac{e^{z}}{z^n}dz}$.
My intuition tells me that I can use the Cauchy Integral Formula since the function, $f(z)$ has $n$ singular points, being $z=0$ which lie entirely in the unit circle.
My solution is as follows:
$${\int_{|z|=1} \dfrac{e^{z}}{z^n}dz}=\int_{|z|=1}\dfrac{e^{z}}{(z_{1}-0)^n}+...+\int_{|z|=1}\dfrac{e^{z}}{(z_{n}-0)^n} =n[2\pi i.f(0)]=2\pi nei,$$ where $f(0)= \dfrac{e^{0}}{(z-0)^0} = 1.$
Can someone tell me if i am on the right track and how can i correct my solution if i am wrong. Many thanks!
You are not using the Cauchy's integral formula Theorem in the correct way. Recall the statement and use it with $a=0$ and $f(z)=e^z$: for any $n \in \mathbb{N}^+$, $$\int_{|z|=1} \dfrac{f(z)}{z^n}dz=2\pi i \cdot\frac{ f^{(n-1)}(0)}{(n-1)!}.$$ For $n=0$ the integral is zero by Cauchy's integral theorem.