Depending on one's approach to Complex Analysis in One Variable, Cauchy's Integral Theorem is one of the first interesting results about holomorphic functions in any course. There are several related versions of it:

  • "Cauchy-Green": let $U\subseteq\mathbb{C}$ be open and simply connected, $f:U\to\mathbb{C}$ holomorphic with continuous derivative $f'(z)$, and $\gamma$ a simple closed smooth contour in $U$, then $\oint_\gamma f(z)dz = 0$. Proof: $\omega:=f(z)dz$ is closed and $C^1$, so one can use Green-Stokes' theorem or the homotopy-invariance of $\omega$, whose all proofs, however, again require $C^1$-regularity to my best knowledge.
  • "Homological Cauchy Integral Theorem": the same argument that $\omega$ is closed and $C^1$ when combined with Stokes - de Rham immediately yields the analogous statement for a nullhomologous $\Gamma\in Z_1(U,\mathbb{Z})$ in a general $U$.
  • "Cauchy-Goursat-Pringsheim": drops the condition for continuity of the derivative by first establishing the claim for triangle contours, from where one can proceed in various ways. However, it doesn't address $1$-cycles.
  • "Cauchy-Artin": proves the statement for null-homologous $1$-cycles while also dropping the need for continuity of $f'(z)$, however needs first to establish a connection between null-homology and winding numbers. In many (if not most) books, probably starting with Ahlfors', the latter issue is rather erroneously swept under the rug by simply declaring that a cycle is called null-homologous iff all its winding numbers with respect to points outside of the given set containing it are $0$. Personally, I prefer to keep both notions apart since the one is of purely topological origin and the other is of analytic nature and equating them should be and is a theorem (due to Artin).

Hence my question:

Is there a reasonable proof of Cauchy's Integral Theorem for $1$-cycles without invoking continuity of the derivative and without invoking winding numbers?

Formally, the answer is "yes, of course, by simply translating all the winding number stuff in Artin's proof back to homology", but I hope it's clear by now it's not what I have in mind as reasonable. The motivation is to be able to present a general Cauchy Integral Theorem as early as Cauchy-Goursat, in particular before introducing $dz/(z-a)$.


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