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 $0$.
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$\frac{1}{2} z^2$ has an antiderivative $F(z)=\frac13 z^3$. Therefore the answer is $F(i)-F(i-1)$.