Complex integral of $\cfrac{e^{2z}}{z^4}$

Question

I want to calculate $\displaystyle\int\cfrac{e^{2z}}{z^4}dz$ where $ |z| \le 1$.

I write $z=x+iy$ then split the integral into real and imaginary parts but could not find a way to parametrize $z$ or to get rid of $e^{2z}$.

Any ideas would be very helpful.

Answer 1

I am assuming the $|z|\leq 1$ should be a $|z|=1$.

In which case we have by Cauchy's integral formula $$ \int_{|z|=1}\frac{e^{2z}}{z^4}\mathrm dz=\frac{2\pi i}{3!}\frac{d^3}{dz^3} e^{2z}\bigg\vert_0\\ =\frac{16\pi i}{6}=\frac{8\pi i}{3} $$ Of course you may also use the Residue theorem for higher order poles if you would like as well.