In the text theory of Functions of a Complex Varible i'm attempting to calculate the following integrals in $(1)$
$(1)$
$$\frac{1}{2 \pi i} \oint_{\Gamma_{2}} \frac{\zeta^{2} + 5 \zeta}{\zeta -2}d \zeta - \frac{1}{2 \pi i} \oint_{\Gamma_{1}} \frac{\zeta^{2} + 5 \zeta}{\zeta - 2}d\zeta$$
$\text{Remark}$:
Let $\Gamma_{1}$ be the curve $\partial D(0,1)$ and let $\Gamma_{2}$ the curve $\partial D(0,3)$ both be equipped with counterclockwise orientation. The two curves from the boundary of an annulus $\Phi$. Rigsourly speaking $\Psi \subset \Phi \subset \mathbb{C}$, note $\Psi$ is our open subset.
Our integrad: $\frac{\zeta^{2} + 5 \zeta}{\zeta -2}$, can be factored as follows and one gets the developments in $(2)$
$(2)$
$$\frac{1}{2 \pi i} \oint_{r_{1} < |z-0| < 1} \frac{ \zeta( \zeta^{} + 5 \zeta)}{\zeta -2}d \zeta - \oint_{r_{1} < |z-0| < 3}\frac{ \zeta( \zeta^{} + 5 \zeta)}{\zeta -2} d \zeta$$
In order to achieve calculations for our integrals in $(2)$, one will need to represent $f(z)=\frac{ \zeta( \zeta^{} + 5 \zeta)}{\zeta -2}$ as a Laurent series as formally discussed in $\text{Lemma (1.2)}$
$\text{Lemma (1.2)}$
$$\text{Theorem 7.19} \, \, \, \text{(Laurent Series)}$$
If $f(z)$ is analytic throughout the annular region: $A: r_{1} < |z-z_{o}| < r_{2}$ there is a series expansion as follows in $(1)$
$(1)$
$$f(z) = \sum_{}^{}a_{k}(z-a)^{k} + \sum_{}^{}b_{k}(z-a)^{-k}$$
where,
$$a_{k}= \frac{1}{2 \pi i} \oint_{\Gamma}\frac{f(\zeta)d \zeta}{(\zeta - a)^{k+1}}$$ $$b_{k} = \frac{1}{2 \pi i } \oint_{\Gamma}(\zeta - a)^{k - 1}f(\zeta)d \zeta$$
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, $ $\text{Remark}:\Gamma \, \text{is any circle} |z-z_{o}| = r \, \text{inside} \, \Phi$
Applying $\text{Theorem 7.19}$, to $(2)$ one can notion the following results in $(3)$
$(3)$
$$\sum \frac{1}{2 \pi i}\oint_{r_{1} < |z-0| < 1} \frac{ \zeta( \zeta^{} + 5 \zeta)d \zeta}{(\zeta -2)^{k+1}}(z-a)^k + \sum \frac{1}{2 \pi i } \oint_{r_{1} < |z-0| < 1}(\zeta -z)^{k+1}\zeta(\zeta + 5 \zeta) - \sum \frac{1}{2 \pi i}\oint_{r_{1} < |z-0| < 3} \frac{ \zeta( \zeta^{} + 5 \zeta)d \zeta}{(\zeta -2)^{k+1}}(z-a)^k + \sum \frac{1}{2 \pi i } \oint_{r_{1} < |z-0| < 3}(\zeta -z)^{k+1}\zeta(\zeta + 5 \zeta)$$
Concerning the recent developments in $(3)$, may I have hints on how to attack $(3)$ ?
My take would be to use residues: reverse the orientation of $\Gamma_1$ and regroup the integrals. Now consider the integral of that same function over the arc $\sigma(t) = 1+2t$ for $t \in [0,1]$. Let $\gamma = \Gamma_2 + \sigma - \Gamma_1 - \sigma$ (changing the starting points of your curves, but who cares). This is an integral over a close path, thus we have: $$\frac{1}{2\pi i} \int_\gamma \frac{\zeta^2-5\zeta}{\zeta-2}d\zeta = Res_{\zeta=2}\frac{\zeta^2-5\zeta}{\zeta-2} = 14$$