Cauchy integral formula

The formulas.

Let D={z:|z-z0|<R} be an open disk in the complex plane, and let f(z) be a holomorphic11It is necessary to draw a distinction between holomorphic functions (those having a complex derivativeMathworldPlanetmath) and analytic functionsMathworldPlanetmath (those representable by power seriesMathworldPlanetmath). The two concepts are, in fact, equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, but the standard proof of this fact uses the Cauchy Integral FormulaPlanetmathPlanetmath with the (apparently) weaker holomorphicity hypothesisMathworldPlanetmath. function defined on some open domain that contains D and its boundary. Then, for every zD we have

f(z) = 12πiCf(ζ)ζ-z𝑑ζ
f(z) = 12πiCf(ζ)(ζ-z)2𝑑ζ
f(n)(z) = n!2πiCf(ζ)(ζ-z)n+1𝑑ζ

Here C=D is the corresponding circular boundary contour, oriented counterclockwise, with the most obvious parameterization given by



The first of the above formulasMathworldPlanetmathPlanetmath underscores the “rigidity” of holomorphic functions. Indeed, the values of the holomorphic function inside a disk D are completely specified by its values on the boundary of the disk. The second formula is useful, because it gives the derivativePlanetmathPlanetmath in terms of an integral, rather than as the outcome of a limit process.


The following technical generalizationPlanetmathPlanetmath of the formula is needed for the treatment of removable singularities. Let S be a finite subset of D, and suppose that f(z) is holomorphic for all zS, but also that f(z) is boundedPlanetmathPlanetmathPlanetmath near all zS. Then, the above formulas are valid for all zDS.

Using the Cauchy residue theorem, one can further generalize the integral formula to the situation where D is any domain and C is any closed rectifiable curve in D; in this case, the formula becomes


where η(C,z) denotes the winding number of C. It is valid for all points zDS which are not on the curve C.

Title Cauchy integral formula
Canonical name CauchyIntegralFormula
Date of creation 2013-03-22 12:04:46
Last modified on 2013-03-22 12:04:46
Owner djao (24)
Last modified by djao (24)
Numerical id 25
Author djao (24)
Entry type Theorem
Classification msc 30E20