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Hey everyone so I have a quick question about this integral symbol: $$\oint$$ I know that its referred to as a closed line integral for real functions usually like this: $$\oint_{C} $$or a closed contour integral for complex functions usually like this: $$\oint_{γ} $$ However is there a general name for this symbol? I heard that CLOSED CONTOUR INTEGRAL or PATH INTEGRAL are the names that cover each occasion but I am not sure. Also if it helps in greek this is referred to as the following: Επικαμπύλιο Ολοκλήρωμα meaning the integral over a curve.How could this be translated over to English?

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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic
    Jan 17, 2023 at 20:38
  • $\begingroup$ See this. As far as I am aware it does not have a specific name other than a "Contour/Path integral". The subscript symbol is just the name of the contour we're assessing around, I wouldn't worry about using certain letters for different types of functions. $\endgroup$
    – B-Foster16
    Jan 17, 2023 at 21:49
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    $\begingroup$ "Line/path/contour integral" on their own are mutually synonymous, but "contour" tends to be associated more with integrals in the complex plane (anecdotally, anyway). "Path integral" seems to have another special meaning in quantum mechanics. $\endgroup$
    – user170231
    Jan 17, 2023 at 22:46

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According to Wikipedia, the terms "line integrals", "path integrals", and "curve integrals" ("line integrals" are by far more adopted) are usually used in the context of vector Calculus, while "contour integrals" are restricted to the context of integrals in the complex plane.

Indeed, I always read the term "contour integral" in the context of complex analysis. However, I never read an authoritative source the exactly state that.

It is not true that

$$ \oint_C $$

is restricted for closed line integrals of real-valued functions. The inverse z-transform, for instance, is frequently denoted with $\oint_C$. That is right that $\oint_\gamma$ is also used, sometimes. But it seems to be a question of author taste. Not big deal.

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It depends on the context. If you are integrating in multivariable calculus then it is usually called line integral, although at times it is called a contour integral. The same thing is true in complex analysis. But path integral isn't used, and according to @user170231 it is something used in quantum mechanics.

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