Let us consider the complex integral $$ \int_{\partial B_r(0) \subseteq \mathbb{C}} \frac{1}{z-1}~\mathrm{d}z. $$ Of course, it vanishes if $r < 1$. It is $2\pi i$ if $r>1$ (see e.g. Cauchy-Integral-Formula). But if $r=1$, neither the Residue Theorem, nor the Cauchy-Integral-Formula can be applied. So how can I approach it then and is it well-defined?
And in general: How to tackle integrals $$ \int_{\partial B_r(0)} \frac{1}{g(z)}~\mathrm{d}z $$ if $z \mapsto \frac{1}{g(z)}$ is holomorphic but has a singularity in $\partial B_r(0)$? Are there some particularly nasty examples for this?
As an improper integral, it is not defined. However, the Cauchy Principal Value (the limit as $\epsilon \to 0$ of the integral from $e^{i\epsilon}$ to $e^{i(2\pi-\epsilon)}$) does exist and is $i\pi$.