Given are the integration paths $\alpha, \beta, \gamma: [0,1]\to \mathbb{C}$ and $ \delta : [0,3]\to\mathbb{C}$: $$\begin{align*} \alpha(t)&=2,5e^{2\pi i t}\\ \beta(t)&=-1,5i+1,5\cos(\pi(t+1))+0,5 i \sin(\pi(t+1)) \\ \gamma(t)&=-1,5i+2it \\ \\ \delta(t)&= \left\{ \begin{array}{ll} -1+0,5e^{i\pi(1/2-2t)} &, 0\leq t \leq 1\\ -1+0,5i+2(t-1) &, 1\leq t\leq 2\\ 1+0,5e^{i\pi(9/2-2t)} &, 2\leq t\leq 3 \end{array}\right.\\ \end{align*} $$ Sketch the trace of the chain $\Gamma=\alpha+\beta+\gamma+\delta$ and compute $$\displaystyle \int_{\delta} z\, \mathrm{d}z$$.
First I want to make this equations easier: $$ \beta(t)=-1,5i -1,5 e^{\frac{1}{3}i\pi t}\\ \gamma(t)= 2i\left(t-\frac{3}{4}\right)$$ Only for $\delta(t)$ I didn't get something useful.
Unfortunately I don't know how to sketch this. To find the starting and ending point of each path, maybe I have to calculate $\alpha(0), \alpha(1)$, etc. Is this the right way? Thank you!
Hint. Note that in the complex plane,
path $\alpha$ is a circle centered at $0$ of radius $2.5$.
path $\beta$ is a half of an ellipse centered at $-1.5i$, semi-major axis of $1.5$ and semi minor axis of $1.5$.
path $\gamma$ is a segment from $-1.5i$ to $0.5i$.
Are you able to sketch the path $\delta$ now?
As regards the integral $\int_{\delta}z\,dz$, note that it suffices to find the endpoints of the path $\delta$ (why?).