# 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 $\mathbb{R}^2$? – Ron Gordon 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. – Cameron Williams 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 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. – Jonas Meyer 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. – dustin 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. – dustin 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. – André 3000 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. – Cameron Williams Dec 24 '14 at 20:39