I don't know where to begin with this question. Can anyone help me?
Let be a semicircular path with centre at and with radius of 1 in the left half plane, that is, it is the straight line from 2 to 0 followed by left half a circle from 0 back to 2i.
Evaluate $$\oint_C \bar z dz$$
hint
Along the vertical line, we have
$$z=it$$ with $$0\le t\le 2, \bar{z}=-it, \;\; dz=idt$$
along the half-circle
$$z-i=e^{it}$$
with $$\frac{\pi}{2}\le t\le \frac{3\pi}{2}$$