Integrate counterclockwise around the unit circle.
$$\int_c \frac{e^{z}}{z^{n}}dz$$ where n = 1,2,...
Where do I even begin this?
I know the integral formula that I probably want to use is:
$$\int \frac{f(z)}{(z-a)^{n+1}}dz = \frac{2\pi \cdot i}{n!} \cdot f^{n}(a)$$
Is the answer just going to be in general form?
So $e^{z}$ derivatives are just $e^{z}$. What should I do with z? Do I have to convert it?
That's it!
$$\oint_C \frac{e^z}{z^n} dz = \frac{2\pi i}{(n-1)!} \left. \frac{d^{n-1}}{dz^{n-1} }e^z \right |_{z=0}$$
$$\oint_C \frac{e^z}{z^n} dz = \frac{2\pi i}{(n-1)!}$$
Another way $\cdots$
$$\oint_C \frac{e^z}{z^n} dz = \oint \frac{dz}{z^n}\left[1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots \right]=2\pi i\oint \frac{dz}{z^n}\frac{z^{n-1}}{(n-1)!}=\frac{2\pi i} {(n-1)!},$$ since all other terms evaluate to zero.
