Cauchy integral theorem

2020 Mathematics Subject Classification: Primary: 30-XX Secondary: 32-XX [MSN][ZBL]

A fundamental theorem in complex analysis which states the following.

Theorem 1 If $D\subset \mathbb C$ is a simply connected open set and $f:D\to \mathbb C$ a holomorphic funcion, then the integral of $f(z)\, dz$ along any closed rectifiable curve $\gamma\subset D$ vanishes: \begin{equation}\label{e:integral_vanishes} \int_\gamma f(z)\, dz = 0\, . \end{equation}

The left hand side of \eqref{e:integral_vanishes} is the integral of the (complex) differential form $f(z)\, dz$ (see also Integration on manifolds). More precisely, if $\alpha: \mathbb S^1 \to \mathbb C$ is a Lipschitz parametrization of the curve $\gamma$, then \begin{equation}\label{e:formula_integral} \int_\gamma f(z)\, dz = \int_0^{2\pi} f (\alpha (t))\, \dot{\alpha} (t)\, dt\, \end{equation} (observe that in order for \eqref{e:formula_integral} to be well defined, i.e. independent of the chosen parametrization, we must in general decide an orientation for the curve $\gamma$; however since \eqref{e:integral_vanishes} stipulates that the integral vanishes, the choice of the orientation is not important in the present context).

An equivalent version of Cauchy's integral theorem states that (under the same assuptions of Theorem 1), given any (rectifiable) path $\eta:[0,1]\to D$ the integral \[ \int_\eta f(z)\, dz \] depends only upon the two endpoints $\eta (0)$ and $\eta(1)$, and hence it is independent of the choice of the path of integration $\eta$. This, essentially, was the original formulation of the theorem as proposed by A.L. Cauchy (1825) (see [Ca]); similar formulations may be found in the letters of C.F. Gauss (1811). Cauchy's proof involved the additional assumption that the (complex) derivative $f'$ is continuous; the first complete proof was given by E. Goursat [Go2]. The property of analytic functions expressed by the Cauchy integral theorem fully characterizes them (see Morera theorem), and therefore all the fundamental properties of analytic functions may be inferred from the Cauchy integral theorem.

For an arbitrary open set $D\subset \mathbb C$ or on a Riemann surface, the Cauchy integral theorem may be stated as follows: if $f:D\to \mathbb C$ is holomorphic and $\gamma \subset D$ a closed rectifiable curve homotopic to $0$, then \eqref{e:integral_vanishes} holds.

A generalization of the Cauchy integral theorem to holomorphic functions of several complex variables (see Analytic function for the definition) is the Cauchy-Poincaré theorem.

Theorem 2 If $D\subset \mathbb C^n$ is an open set and $f:D \to \mathbb C$ a holomorphic function, then for any smooth oriented $n+1$-dimensional (real) surface $\Sigma$ with smooth boundary $\partial \Sigma$ we have \[ \int_{\partial \Sigma} f(z)\, dz = 0\, , \] where $dz$ denotes the differential form $dz_1\wedge dz_2 \wedge \ldots \wedge dz_n$.

When $n=1$ the surface $\Sigma$ and the domain $D$ have the same (real) dimension (the case of the classical Cauchy integral theorem); when $n>1$, $\Sigma$ has strictly lower dimension than $D$. See also Residue of an analytic function; Cauchy integral.

References

[Al]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904
[Ca]	A.L. Cauchy, "Oeuvres complètes, Ser. 1" , 4 , Paris (1890)
[Go]	E. Goursat, "Démonstration du théorème de Cauchy" Acta Math. , 4 (1884) pp. 197–200
[Go2]	E. Goursat, "Sur la définition générale des fonctions analytiques, d'après Cauchy" Trans. Amer. Math. Soc. , 1 (1900) pp. 14–16
[Ma]	A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[Sh]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[Vl]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)