Question 5

Let $${\alpha = \frac{dz}{z + 1}}$$.

• Discuss in which regions in the complex plane, the differential $\alpha$ is exact.

Sketch and parametrize each of the following curves and hence, or otherwise, compute the integral of $\alpha$ along each curve.

• The semi-circular arc centered at the origin of radius $1$ starting at $z = - i$ and ending at $z = i$, with $\Re(z) \ge 0$, traced counter-clockwise.
• The arc of the ellipse with Cartesian equation $${\frac{x^2}{4} + y^2 = 1}$$, starting at $z = i$ and ending at $z = - i$, with $\Re(z) \le 0$, traced counter-clockwise.
• The straight line from $z = i$ to $z = -2$ followed by the straight line from $z = -2$ to $z = - i$, followed by the straight-line from $z = - i$ to $z = i$.