Use this parametrization to compute the following integral.

Question

Let $$\Gamma$$ be the circumference centered at 1-i of radius 5 and transversed once in the counterclockwise direction. Parametrize the contour $$\Gamma$$. Use this parametrization to compute the following integral: $$\int \frac{1}{z-1+i}$$

I am hoping to see a solution to this problem, as this entire unit has been confusing for me.

Answer 1

Since $z\in \Gamma$ if and only if $|z - (1 - i)| = 5$, we can parametrize $\Gamma$ by setting $z = 1 - i + 5e^{it}$, $0 \le t \le 2\pi$. Then $dz = 5ie^{it}\, dt$ and so

$$\int_{\Gamma} \frac{dz}{z - 1 + i} = \int_0^{2\pi} \frac{5ie^{it}}{5e^{it}}\, dt = i\int_0^{2\pi} dt = 2\pi i.$$

Answer 2

$c(t)=(1-i)+5e^{it}$

then

$\int_\Gamma\dfrac{5ie^{it}}{5e^{it}}dt=\int_o^{2\pi}idt=2\pi i$