1) Let $\gamma_1$ be the straight line segment from $-1+i$ to $0$ and let $\gamma$ be the straight line segment from $0$ to $i$. Let $\gamma$ be the concatenation of $\gamma_1$ and $\gamma_2$. Calculate
$\int_\gamma z^2 dz.$
2) Let $P$ be the regular polyon whose vertices are the fifth roots of $7 + 8i$, orientated counterclockwise. Calculate
$\int_P \frac{dz}{z^2(z-3i)}$
3) Calculate all the possible values of
$\int_P \frac{dz}{(z-4)(z-3i)}$
where $\gamma$ denotes any simple closed curve $\gamma$ $\mathbb{C}$, orientated counterclockwise.
The answers I got are -
1) $-\frac{i}{2} + \frac{2}{3}\sqrt2e^{i\frac{\pi}{4}}$
2) The vertices of the polygon lie on the circle with radius = 10th root of 113. This is approx = 1.6 so the only singularity inside the polygon is z = 0. So the answer is $-\frac{2\pi}{3}$
3)
- $\gamma$ encloses neither of the singularities, then by Cauchy - Goursat the integral = 0
- $\gamma$ encloses either of the singularities. Two values: $-\frac{2{\pi}i}{4-3i}$ and $\frac{2{\pi}i}{4-3i}$
- $\gamma$ encloses both curves. The integral is equal to the sum of the integrals in the previous part, so it's equal to $0$.