A simple question: Where did the contour integral sign appear for the first time?

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Wikipedia says that it was introduced by physicist Arnold Sommefield in 1917 ( Table of mathematical symbols by introduction date ) but there are not any references to prove this

Does anyone know if this is true? Is there a scientific paper or a book where this symbol was introduced?

  • $\begingroup$ My suspicion is that the notation was invented by Cauchy as he was the earliest mathematician to theorise the integral with 'complex limits' and Cauchys theorem very definitely uses a contour. Mathematicians being a conservative lot would have simply carried on with notation as such. However, this is my guess. I'd put it as a comment but for the moment my so-called smart phone isn't able to enter comments ... $\endgroup$ Aug 24, 2020 at 2:00

1 Answer 1


Jeff Miller's very valuable collection of the origins of mathematical expressions has the entrie "Integration around a closed path":

Dan Ruttle, a reader of this page, has found a use of the integral symbol with a circle in the middle by Arnold Sommerfeld (1868-1951) in 1917 in Annalen der Physik, "Die Drudesche Dispersionstheorie vom Standpunkte des Bohrschen Modelles und die Konstitution von H2, O2 und N2." This use is earlier than the 1923 use shown by Cajori. Ruttle reports that J. W. Gibbs used only the standard integral sign in his Elements of Vector Analysis (1881-1884), and that and E. B. Wilson used a small circle below the standard integral symbol to denote integration around a closed curve in his Vector Analysis (1901, 1909) and in Advanced Calculus (1911, 1912).

Even if Cauchy actually used contour integrals, he didn't use the symbol: what we know as "Cauchy's integral formula" was proved in Cauchy, Sur les integrales qui s'etendent a tous les points d'une courbe fermee, Camp. Rend., 23 (1846), 251-255, you can see here that Cauchy simply used the standard integration symbol.

Until at least 1905 that symbol was not in use, see this paper of Dimitrie Pompeiu where Cauchy's formula is stated using the standard integration symbol.

  • $\begingroup$ I think you are right, I have already checked Sommerfield's paper, and in page 510 it is possible to see integrals with middle circles, altough probably its inspiration was E. B. Wilson's small circile notation. Thanks $\endgroup$
    – Nau
    Aug 25, 2020 at 19:17

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