Simple Contour Integral

Question

I have forgotten much of the complex analysis I once knew. How do I go about using the Cauchy Integral Formula / Residue Theorem to solve this contour integral? The region is the unit circle.

$$\oint \frac{(1 + z)^4}{z^3} dz.$$

Answer 1

HINT:

$$\frac{(1+z)^4}{z^3}=z^{-3}+4z^{-2}+6z^{-1}+4+z$$

The residue is the coefficient on the $z^{-1}$ term of the Laurent series.

Answer 2

Hint:

$$f^{(n)}(z_0)=\frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z-z_0)^{n+1}}dz$$

And $f(z)$ is analytic.

Answer 3

As suggested by @MrYouMath Using the Cauchy Integral Formula we have$$\oint_{|z|=1} \frac{(1+z)^4}{z^3} dz =\frac{2\pi \iota [12(1+z)^2]_{z=0}}{2!}=12\pi\iota $$