The residue theorem states:
Suppose $U$ is a simply connected open subset of the complex plane, and $a_1,\ldots,a_n$ are finitely many points of $U$ and $f$ is a function which is defined and holomorphic on $U\setminus\{a_1,\ldots,a_n\}$. If $\gamma$ is a rectifiable curve in $U$ which does not meet any of the $a_k$, and whose start point equals its endpoint, then
$$\oint_\gamma f(z) dz = 2\pi i \sum_{k=1}^n I(\gamma, a_k) \textrm{Res}(f, a_k)$$
But what if a pole $a_k$ lies on the curve $\gamma$? What could we say about the evaluation of our integral?
Take for instance,
$$\oint_C \frac{1}{z+1} dz$$
where $C$ is the unit circle. Is it fair to say that the value of this integral is 0 by the residue theorem?
Best Answer
Such an integral simply is not defined. You should change the curve and avoid the poles.
Even if we go down to the basics: $$\int_\gamma f dz = \int_a^b f(\gamma(t)) \gamma'(t) dt$$ But $f(\gamma(t))$ is not defined for the
pole, thus the integral is undefined.Here's a related question with answers: general-method-of-integration-when-poles-on-contour And here's a nice image on how to avoid poles: Residue theorem:When a singularity on the circle (not inside the circle)