Question 1

Let $${\alpha = \overline{z} dz}$$. Find the contour integral of $\alpha$ taken around the following contours each traced once around counter-clockwise:

• $\mathcal{A}$: the circle $\vert z- 5\vert = 5$.
• $\mathcal{B}$: the triangle $CDE$ with vertices at the points:
  $C = 3 + 2i, \hspace{3pt} D = 1 + i$ and $E = 2 -i$.

Is the differential $\alpha$ exact on the region outside the circle $\mathcal{A}$? Explain.