# Why do we have separate complex-integration and contour-integration tags?

It seems like these are nearly identical in nature. The tag for complex integration was created three months ago and the contour-integration tag has been around for over two years. I feel like the complex-integration tag should be a synonym for contour-integration.

• Hmmm. I am inclined to agree, as virtually all instances of contour-integration indicate the complex plane. But is this necessarily the case? How about real line integrals over curves in $\mathbb{R}^2$? Dec 22 '14 at 19:56
• There's a separate line-integrals tag floating around somewhere which would take care of that, I think. Line integral problems don't show up quite as much, so I think it's okay to distinguish contour integration from line integration. Also the wiki for contour-integration specifically has complex plane integration in mind for that matter. Dec 22 '14 at 19:58
• when starting complex analysis, sometimes there are problems asking one to integrate complex functions that are not complex-differentiable over curves in $\mathbb{C}$. These are the only reason I might see for the complex-integration tag.
– robjohn Mod
Dec 22 '14 at 20:01
• @robjohn: I think those cases would still fall under the umbrella of contour integration; the contour integral is defined without regard to differentiability, e.g. via Riemann-Stieltjes integration. Even when concerned primarily with holomorphic functions, one uses contour integrals for functions not a priori known to be holomorphic, e.g. in applying Morera's theorem. Dec 22 '14 at 20:15
• One could integrate over a subset of the plane that is not a contour. However, the description of complex integration tag suggests it is to be used synonymously with contour integration. So I think the tags should be synonyms.
– user147263
Dec 22 '14 at 20:29
• @Behaviour we could easily integrate a complex valued function without using a contour. For instance, take this post, I would call this complex integration but it wouldn't fit in with contour integration. Dec 22 '14 at 21:05
• @dustin Why would not it fit? There's a contour of integration right there, the boundary of the rectangle.
– user147263
Dec 22 '14 at 21:10
• @Behaviour You are correct. I was thinking of something else. I was thinking of it as solely applying to residues. Dec 22 '14 at 21:12
• Wait, do people actually distinguish between contour, line, and path integrals? To me, they are synonymous and make sense for any nice manifold, not just the complex plane. Dec 24 '14 at 19:24
• @SpamIAm I think they only distinguish contour integrals as they show up repeatedly in complex analysis although it's just a realization of a line integral. Just to distinguish the context easily. Dec 24 '14 at 20:39