Given is the following chain $\Gamma=\alpha +\beta+\gamma+\delta$:
$\gamma$ is a half circle around $z_0=4$. Parametrize this paths $\alpha, \beta, \gamma, \delta : [0,1]\to\mathbb{C}$.
I'm not sure how to do this. For $\beta$ one can say it's a constant path, so $$\beta = 2 = 2\cdot e^{2\pi \mathrm{i}t}$$ For $\delta$ I'd say: $$\delta=-2t+2 \, t\in [0,2]$$ and for $$\alpha = \frac{1}{6}t+\frac{1}{3}, \, t\in [-2, 10]$$
For a circle one can write $D_r(z_0)=r\cdot e^{\mathrm{i}t}+z_0$. So I get for a full circle in this situation: $$D_r(4)=r\cdot e^{\mathrm{i}t}+4$$ Unfortunately I don't if these things are correct, especially the half circle. Any hints?
You have only straight line segments from $z_0$ to $z_1$:
$$ z(t)=(1-t)z_0 + t z_1 \quad t \in [0,1] $$
and the arc of a circle with centre $z_0$ and radius $r$: $$ z(t)=z_0 + r ( \cos t + i \sin t), \quad t \in [t_1, t_2] $$ $t_1$ is the initial point of the arc, $t_2$ is the final point, traversed counter-clockwise ( use $-i$ for clockwise traversal. )
You will have to think about the parameter $t$ if you want $t \in [0,1]$ for the collection of paths.