Question 3

Let $${\beta = \frac{dz}{z^2 - 2z + 2}}$$.

Consider the contour integrals of $\beta$ taken around the following contours:

• $\mathcal{D}$: The circle $\vert z\vert = 2$, traced once around counter-clockwise.
• $\mathcal{E}$: The figure of eight contour, which goes first around the circle with equation $\vert z - 2i\vert = 2$ once around counter-clockwise, then around the circle $\vert z + 2i\vert = 2$, once around clockwise.
• $\mathcal{F}$: The square with vertices at $10 + 10i, 10 - 10i, -10 - 10i, -10 + 10i$, traced once around counter-clockwise.
• $\mathcal{G}$: The circle $\vert z - 10i\vert = 10$ traced once around counter-clockwise.
• $\mathcal{H}$: The circle $\vert z + 10i\vert = 10$ traced once around counter-clockwise.

At least one of these integrals is zero. Which? Explain.
How are these various integrals related? Explain.