This is just a plan-out.
I want to evaluate:
$$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$
Using a keyhole contour a semi-circle, with base at the x-axis.
First I must pick a branch.
Since the logarithm is multivalued in that,
$$\log(e^{i2\pi}) = \log(e^{(2ki\pi) + i2\pi})$$
We consider: $f(z) = \displaystyle \frac{\log^2(z)}{z^2 + 1}$
But how should I pick a branch.
(1) What is a "Branch?"
(2) Should I pick a place where the $\log$ is continous. That is everywhere except $z=0$
So then my contour will be half-washer shape. with the opening in the middle: Half a Washer
Thanks