Evaluating a complex integral on a circle

Question

I have the function

$ f(z) = \frac{z^3}{z^2+i} $

and I'm trying to calculate the integral:

$ \int_{C(0;2)} f(z)dz$

where $C$ is the circle centered at the origin with radius $2$.

Could someone explain how I might go about approaching this.

Thanks a lot

Answer 1

The roots of $z^2+i$ are $\pm e^{\frac{i\pi}{4}}$ and they are both in the disk $D(0,2)$. Hence $$\int_{C(0,2)} \frac{z^3}{z^2+i} dz = 2i\pi (Res(f,e^{\frac{i\pi}{4}})+Res(f,-e^{\frac{i\pi}{4}})).$$

EDIT : If you don't know the residue theorem but only the Cauchy's formula, one other way to do it is to use a decomposition $$\frac{z^3}{z^2+i} = \frac{z^2}{2} \left(\frac{1}{z-e^{\frac{i\pi}{4}}} + \frac{1}{z+e^{\frac{i\pi}{4}}}\right)$$ and apply Cauchy's formula twice.