Evaluate $\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$ where $r>0$

Question

I tried to evaluate $$\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$$ when $r>0$. Clearly $\log z$ is not continuous at $z=-r$. So is this integral meaningful then? Since the function is continuous and bounded on the given path except only at this point, I am thinking it should be possible to actually evaluate this. Hope someone could help me out with this confusion and help to evaluate this integral. Thanks

Answer 1

A branch of the logarithm is defined in a region, $G\subseteq\dot{\mathbb{C}}$, if and only if for every cycle, $\Gamma$, contained in $G$, $n(\Gamma, 0) = 0$, where $n(\Gamma,0)$ is the winding number, which represents the number of times the curve goes around $0$.

In your particular case, whichever region containing the curve you're integrating over doesn't satisfy this condition, so you can't define a branch of the logarithm.

Basically, you can't find any branch of the logarithm which is well defined in a region containing your curve.

Note: I've seen in some places the notation $\operatorname{Log}$ used to refer to the principal branch. If your adhering to this convention, then everything is ok. If you aren't then it is possible to define a branch of the logarithm in the negative real axis, as long as the condition I mentioned is satisfied.