Are the solution answers correct ? I would appreciate it if you could confirm, i have submitted the evaluate integral questions and solution workouts in images below.
I am posting on behalf of my friend, English is her second language so i will help to convey to her anything she does not quite understand, maths is not my field however she can interpret any input in a maths sense.
Here are images of the two questions, and solutions, solution 1 is at the top and solution 2 at the bottom.
any input is greatly appreciated!
Many thanks!
Both solutions are wrong.
Concerning the solution to question number 1, it makes no sense to say that the number of zeros is $\frac{2n+1}2$ and that the number of poles is $n$. What is $n$? There is no $n$ in the question.
If $f(z)=\sin(\pi z)$, the $f$ has $7$ zeros ($0$, $\pm1$, $\pm2$, and $\pm3$) when $|z|<\pi$, and $0$ poles. Therefore\begin{align}\int_{|z|=\pi}\pi\cot(\pi z)\,\mathrm dz&=\int_{|z|=\pi}\frac{f'(z)}{f(z)}\,\mathrm dz\\&=2\pi i\times(7-0)\\&=14\pi i.\end{align}
Concerning the solution to question number 2, the answer cannot be $0$, since we are dealing here with the integral of a function which takes only values greater than $0$ (when $z\in\Bbb R$). Note that the zeros of $z^4+4$ are $\pm\sqrt2e^{\pi i/4}$ and $\pm\sqrt2e^{3\pi i/4}$. Of these four, those with imaginary part greater than $0$ are $\sqrt2e^{\pi i/4}$ and $\sqrt2e^{3\pi i/4}$. So, your integral is equal to\begin{align}2\pi i\left(\operatorname{res}\left(\sqrt2e^{\pi i/4},\frac1{z^4+4}\right)+\operatorname{res}\left(\sqrt2e^{3\pi i/4},\frac1{z^4+4}\right)\right)&=2\pi i\left(\frac{-1-i}{16}+\frac{1-i}{16}\right)\\&=\frac\pi4.\end{align}