Question
a) $\int_{\Gamma}z\sin(z^2)dz$, where $\Gamma : \gamma(t):= \sqrt{3}t+2(1-t)i$, where $0 \leq t < 1$.
b) Calculate $\int_{\Gamma} \frac{\sin(z)}{z^2-5}+ \frac{e^-z}{z-2}dz$, where $ \Gamma : |z-12i|= \frac{1}{4} $.
My Answer
a) I don't know how I should start this part. I have tried differentiating $\gamma(t)$ and using the formula $\int_{0}^{1}(\gamma(t)\sin(\gamma(t)^2)\gamma'(t)dt$ but this becomes very complicated. I have also tried using $\int_{\Gamma} f(z)dz=F(q)-F(p)$, where $p$ and $q$ are the initial and final points of $\Gamma$ but this didn't work either.
b) For this part I think you can use Cauchy's integral formula but don't know how to do this. Would I need to make $f(z)$ into one function and then calculate the residues for the poles that are within the interior of $\Gamma$? I also just wanted to check is $\Gamma$ a circle centre $12i$ and radius $\frac{1}{4}$?