Cauchy integral theorem

Theorem 1

Let $U\mathrm{\subset }\mathrm{C}$ be an open, simply connected domain, and let$f\mathrm{:}U\mathrm{\to }\mathrm{C}$ be a function whose complex derivative, that is

$$\underset{w\to z}{lim}\frac{f(w)-f(z)}{w-z},$$

exists for all $z\mathrm{\in }U$.Then, the integral (http://planetmath.org/Integral2) around every closed contour$\gamma \mathrm{\subset }U$ vanishes; in symbols

$${\oint }_{\gamma }f(z)𝑑z=0.$$

We also have the following, technically important generalizationinvolving removable singularities.

Theorem 2

Let $U\mathrm{\subset }\mathrm{C}$ be an open, simply connected domain, and$S\mathrm{\subset }U$ a finite subset. Let $f\mathrm{:}U\mathrm{\backslash \backslash }S\mathrm{\to }\mathrm{C}$ be a function whose complex derivative exists for all $z\mathrm{\in }U\mathrm{\backslash \backslash }S$, and that is bounded near all $z\mathrm{\in }S$. Let $\gamma \mathrm{\subset }U\mathrm{\backslash \backslash }S$ be a closed contourthat avoids the exceptional points. Then, the integral of $f$ around $\gamma$ vanishes.

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

The original version of the theorem, as stated by Cauchy in the early1800s, requires that the derivative $f^{\prime }(z)$ exist and be continuous.The existence of $f^{\prime }(z)$ implies the Cauchy-Riemann equations, whichin turn can be restated as the fact that the complex-valueddifferential $f(z)dz$ is closed. The original proof makes use ofthis fact, and calls on Green’s Theorem to conclude that the contourintegral vanishes. The proof of Green’s theorem, however, involves aninterchange of order in a double integral, and this can only bejustified if the integrand, which involves the real and imaginaryparts of $f^{\prime }(z)$, is assumed to be continuous. To this date, manyauthors prove the theorem this way, but erroneously fail to mentionthe continuity assumption.

In the latter part of the ${19}^{\text{th}}$ century E. Goursat found a proofof the integral theorem that merely required that $f^{\prime }(z)$ exist.Continuity of the derivative, as well as the existence of all higherderivatives, then follows as a consequence of the Cauchy integralformula. Not only is Goursat’s version a sharper result, but it isalso more elementary and self-contained, in that sense that it is doesnot require Green’s theorem. Goursat’s argument makes use ofrectangular contour (many authors use triangles though), but theextension to an arbitrary simply-connected domain is relativelystraight-forward.

Theorem 3 (Goursat)

Let $U$ be an open domain containing a rectangle

$$R=\{x+iy\in ℂ:a\le x\le b,c\le y\le d\}.$$

If the complex derivative of a function $f\mathrm{:}U\mathrm{\to }\mathrm{C}$ existsat all points of $U$, then the contour integral of $f$ around theboundary of $R$ vanishes; in symbols

$${\oint }_{\partial R}f(z)𝑑z=0.$$

Bibliography.

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