# Are contours allowed to contain branch points / branch cuts on the contour (not inside)

I just wanted to verify something, since I saw some notes online that confused me. They were doing some contour integral and $$0$$ was the only branch point of the function, and they chose the branch cut connecting $$0$$ to $${+\infty}$$ on the real axis. The contour they chose was a semi-circular contour. The part that confused me, was that at $$0$$ rather than just "go through" $$0$$ - they made some small $$\epsilon$$ semi-circle around $$0$$, but then allowed themselves to integrate along the branch cut. I thought that as long as the function is analytic at $$0$$ (which it was) - it's fine to integrate over a contour containing branch points/cuts on the contour? So long as you don't actually "cross" that branch cut or encircle any problematic branch points. Am I correct? Or is there a valid reason they did this? Thank you!

As for a path that goes through a branch point, as long as the function has a finite limit as you approach the branch point, there should be no problem. Again you could avoid the branch point by taking a small detour around it, but in the limit as the size of the detour goes to $$0$$ you will get the same result.
• I see that makes sense to me. So integrating along a branch cut isn't a problem, but because of ambiguity in which "side" you are talking about it helps to instead take a contour "hovering" just above or below it and then take a limit? And with the branch point part - with the whole "detour" part, this is exactly what I was thinking - as long as it's nicely behaved at the branch point, any detour will give $0$ in the limit so I thought it seemed pointless. Thank you so much for taking the time to respond :) Feb 23, 2021 at 18:23