$$\oint_{C(-i,1)} \left(\frac {1} {(z+i)^3} - \frac {5} {z+i} +8\right)\,dz$$
The following integral is causing me lots of issues.
I believe Cauchy Integral Theorem applys here and that the solution should be of the form
$I = 2\pi i f(i)$ where $f(z)$ is the numerator of the integrand.
If someone help me with this question I’d really appreciate it. I am really not sure if it is entire or whether Cauchys integral theorem holds.
Perhaps there are better methods to approaching this.
We could rely on the residue theorem and proceed. But, given the form of the integrand and the integration contour, we choose to evaluate the integral directly.
We can parameterize the contour as $z=-i+e^{i\phi}$, $0<\phi\le 2\pi$. Then, we have $$\begin{align} \oint_{|z+i|=1}\left(\frac{1}{(z+i)^3}-\frac{5}{z+i}+8\right)\,dz&=\int_0^{2\pi}\left(\frac1{e^{i3\phi}}-\frac5{e^{i\phi}}+8\right)\,ie^{i\phi}\,d\phi\\\\ &=-10\pi i \end{align}$$