In my studies, I've been uniquely using the Residue Theorem with no more than a single winding around each singularity. Actually, my professor has never mentioned winding-numbers in the Residue Theorem.

I've been reading discussions about complex integrals, where some people suggest solutions that take winding numbers into account. Is there a general advantage in doing so?


migrated from mathoverflow.net Jan 31 at 18:02

This question came from our site for professional mathematicians.


I taught a basic complex analysis class 16 times so far. Initially I did not include the winding number, and restricted the residue theorem to simple closed curves. This is visually satisfactory and usually sufficient for applications, but it gives a hard time for the teacher and the careful student alike. The reason is that one needs to talk about "the interior" and "counterclockwise orientation" of a simple closed curve. However, I found that discussing these concepts rigorously is beyond an introductory course. So, after a while, I decided to include the winding number: it simplified the proofs and made the theorem more general. Here is the version I currently teach in my course:

Residue Theorem. Let $M\subset\mathbb{C}$ be open. Let $\gamma\subset M$ be a closed curve freely homotopic to a constant curve within $M$. Assume that $f$ is holomorphic on $M\setminus S$, where $S\subset M$ is a discrete subset of isolated singularities of $f$ disjoint from $\gamma$. Then $$\int_\gamma f=2\pi i\sum_{p\in S} w_p(\gamma)\mathrm{res}_p(f),$$ where the winding number $w_p(\gamma)$ vanishes for all but finitely many points $p\in S$.

  • $\begingroup$ What does freely homotopic mean? $\endgroup$ – Vít Tuček Jan 31 at 17:02
  • $\begingroup$ @VítTuček: see en.wikipedia.org/wiki/Free_loop $\endgroup$ – Ben McKay Jan 31 at 17:17
  • $\begingroup$ @VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point". $\endgroup$ – GH from MO Jan 31 at 17:41
  • $\begingroup$ That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem. $\endgroup$ – Minamoto Yoshitsune Feb 1 at 16:20

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.