Cauchy's integral formula

from Wikipedia, the free encyclopedia

The Cauchy integral formula (after Augustin Louis Cauchy ) is one of the fundamental statements of function theory , a branch of mathematics . In its weakest form, it says that the values ​​of a holomorphic function inside a circular disk are already determined by their values ​​on the edge of this circular disk. A strong generalization of this is the residual theorem .

Cauchy's integral formula for circular disks


Is open, holomorphic, a point in and a relatively compact circular disk in , then applies to all , i.e. to all with :

It is the positively oriented curve for over the top of .


For solid the function is defined by for and for . is steady on and holomorphic on . With Cauchy's integral theorem, we now have


The function , is holomorphic with the derivative , which vanishes because the integrand has an antiderivative (namely ). So is constant, and because of is .


The following applies to every holomorphic function: The function value in the center of a circle is the mean value of the function values ​​on the edge of the circle. Use it .

Every holomorphic function can be complexly differentiated as often as desired and each of these derivatives is holomorphic. Expressed with the integral formula this means for and :

Every holomorphic function is locally expandable into a power series for .

With the integral formula for it immediately follows that the coefficients are exactly the Taylor coefficients . For the coefficients following estimate applies if, for the following applies:

The set of Liouville (each in a very holomorphic bounded function is constant) can very quickly show the integral formula. With this one can easily prove the fundamental theorem of algebra (every polynomial splits into linear factors).


Cauchy's integral formula is partially differentiated, whereby differentiation and integration can be exchanged:

Development of in Cauchy's integral formula with the help of the geometric series results

Since converges uniformly for the geometric series, one can integrate term by term, i.e. H. Swap sum and integral. The expansion coefficients are:

The following estimate applies to the coefficients. There is a with for ; then applies to :

Is completely holomorphic and restricted, i.e. for all , then as before for all :

Since was arbitrary, then applies to all . Thus it follows from the limitedness of :

That is, every function that is bounded to be entirely holomorphic is constant ( Liouville's theorem ).


Integrals can also be calculated using the integral formula:

Cauchy's integral formula for poly cylinders

Cauchy's integral formula was also generalized to the multi-dimensional, complex space . If circular disks are in , then a poly cylinder is in . Let be a holomorphic function and then Cauchy's integral formula is through

explained. Since Cauchy's integral theorem does not hold in multi-dimensional space, this formula cannot be derived from it analogously to the one-dimensional case. This integral formula is therefore derived from Cauchy's integral formula for circular disks with the help of induction . Using the multi-index notation , the formula can be restored to


with shortened. The formula also applies to multidimensional

for the derivatives of the holomorphic function as well as the Cauchy's inequality

where and is the radius of the poly cylinder . Another generalization of this integral formula is the Bochner-Martinelli formula .

Cauchy's integral formula for cycles

The version for cycles represents a generalization of the integral formula for circular curves:

If a region is holomorphic and a zero-homologous cycle in , then the following integral formula applies to all that are not on :

Here referred to the winding number of order .

Individual evidence

  1. ^ Lars Hörmander : An Introduction to Complex Analysis in Several Variables. North Holland Pub. Co. et al., Amsterdam et al. 1973, ISBN 0-444-10523-9 , pp. 25-27.


  • Kurt Endl, Wolfgang Luh : Analysis. Volume 3: Function Theory, Differential Equations. 6th revised edition. Aula-Verlag, Wiesbaden 1987, ISBN 3-89104-456-9 , p. 153, sentence 4.9.1.
  • Wolfgang Fischer, Ingo Lieb : Function theory. 7th improved edition. Vieweg, Braunschweig et al. 1994, ISBN 3-528-67247-1 , p. 60, chapter 3, sentence 2.2 ( Vieweg study. Advanced course in mathematics 47).