I'm a bit stuck on understanding how to solve a contour integral:

Let $\gamma$ be the circle of radius $1$ centered at $2$ traveled once counterclockwise. Evaluate $\int_\gamma \frac{1}{z^2 - 2z} dz$.

From what I understand, I'm supposed to parameterize this curve, but I don't quite see how. In terms of the bounds, since our curve is a circle of radius $1$ traveled once clockwise, is it simply from $0$ to $2\pi$? Any guidance on this problem would be very helpful. Thank you.


1 Answer 1


There are several ways to solve this problem.

A shortcut is making use of Cauchy’s integral formula(you may want to look it up on Wikipedia).

Link: https://en.m.wikipedia.org/wiki/Cauchy's_integral_formula

In this case. $a=2$, $f(z)=\frac{1}{z}$.

Therefore, the required integral equals: $$2\pi if(a)=\pi i$$


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