$$\int_C \frac{z^2}{z^2 +4} dz$$ where $C$ is the rectangle with vertices$ −2,2,−2 + 4i,2 + 4i$ traversed in the anticlockwise direction.
Anyone know how to do this question? i've tried using cauchy integral formula but i'm not sure on how to do this particular integral as it has a rectangular path.
Best Answer
Within the rectangle, there is only one singularity at $\,z=2i$, thus suppose $\,\gamma\,$ is a small circle around $\,2i\,$, then by $\textbf{Cauchy's integral Formula}$, we have:
$$\int_C\frac{z^2}{z^2+4}\,dz\ =\ \int_\gamma\ \frac{z^2\left/(z+2i)\right.}{z-2i}\,dz\ =\ \left.2\pi i\frac{z^2}{z+2i}\,\right|_{\ z=2i}\ =\ -2\pi$$