Cauchy integral theorem

Theorem 1

Let UC be an open, simply connected domain, and let f:UC be a function whose complex derivativeMathworldPlanetmath, that is


exists for all zU. Then, the integral ( around every closed contour γU vanishes; in symbols


We also have the following, technically important generalizationPlanetmathPlanetmath involving removable singularities.

Theorem 2

Let UC be an open, simply connected domain, and SU a finite subset. Let f:U\SC be a function whose complex derivative exists for all zU\S, and that is bounded near all zS. Let γU\S be a closed contour that avoids the exceptional points. Then, the integral of f around γ vanishes.

Cauchy’s theorem is an essential stepping stone in the theory of complex analysis. It is required for the proof of the Cauchy integral formulaPlanetmathPlanetmath, which in turn is required for the proof that the existence of a complex derivative implies a power seriesMathworldPlanetmath representation.

The original version of the theorem, as stated by Cauchy in the early 1800s, requires that the derivativePlanetmathPlanetmath f(z) exist and be continuousMathworldPlanetmathPlanetmath. The existence of f(z) implies the Cauchy-Riemann equationsMathworldPlanetmath, which in turn can be restated as the fact that the complex-valued differentialMathworldPlanetmath f(z)dz is closed. The original proof makes use of this fact, and calls on Green’s Theorem to conclude that the contour integral vanishes. The proof of Green’s theorem, however, involves an interchange of order in a double integral, and this can only be justified if the integrand, which involves the real and imaginary parts of f(z), is assumed to be continuous. To this date, many authors prove the theorem this way, but erroneously fail to mention the continuity assumptionPlanetmathPlanetmath.

In the latter part of the 19th century E. Goursat found a proof of the integral theorem that merely required that f(z) exist. Continuity of the derivative, as well as the existence of all higher derivatives, then follows as a consequence of the Cauchy integral formula. Not only is Goursat’s version a sharper result, but it is also more elementary and self-contained, in that sense that it is does not require Green’s theorem. Goursat’s argumentMathworldPlanetmath makes use of rectangular contour (many authors use triangles though), but the extensionPlanetmathPlanetmathPlanetmath to an arbitrary simply-connected domain is relatively straight-forward.

Theorem 3 (Goursat)

Let U be an open domain containing a rectangle


If the complex derivative of a function f:UC exists at all points of U, then the contour integral of f around the boundary of R vanishes; in symbols



  • Ahlfors, L., Complex Analysis. McGraw-Hill, 1966.

Title Cauchy integral theorem
Canonical name CauchyIntegralTheorem
Date of creation 2013-03-22 12:54:12
Last modified on 2013-03-22 12:54:12
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 16
Author rmilson (146)
Entry type Theorem
Classification msc 30E20
Synonym fundamental theorem of complex analysis
Related topic ClosedCurveTheorem
Related topic CauchyResidueTheorem
Defines Goursat’s Theorem