I have the following question:
Evaluate $$\int_{\gamma}\frac{z^2+2z}{z^2+4}dz$$ where the contour $\gamma$ is
- The circle of radius $2$ centered at $2i$, traversed once anti-clockwise.
So we have $\gamma(t)=2 e^{it}+2i$.
I am pretty sure I am required to use the Estimation lemma, I've seen similar questions asked but none really which explicitly make use of the Estimation Lemma.
I have my own attempt below:
We can note that $\gamma_R(t)=Re^{it}$ for $R>2$ and $f(z)$ is differentiable on $D=\mathbb C/ (\pm2i)$
$\bigg\vert\int_{\gamma_R}\frac{z^2+2z}{z^2+4}dz \bigg\vert \leq \bigg\vert\int_{\gamma_R}\frac{1}{z^2+4}dz\bigg\vert \leq \int_{\gamma_R}\frac{1}{\vert z^2+4z\vert}dz$
$ \leq max_{z \in \gamma_R}\bigg(\frac{1}{\vert z\vert^2-4}\bigg)L_{\gamma_R} = \frac {1}{R^2-4}L_{\gamma_R}=\frac{2\pi R}{R^2-4} \rightarrow 0$ as $R\rightarrow\infty $
So I attempted this following an alternative example I was shown, however I don't feel like the steps are as applicable here.. I'm not entirely sure. I have seen others attempting with partial fractions but I am struggling to put the Estimation Lemma in practise really so any help would be appreciated.