Cauchy's integral formula
The Cauchy integral formula (after Augustin Louis Cauchy ) is one of the fundamental statements of the theory of functions, of a part of mathematics. It says in its weakest form, that the values of a holomorphic function inside a circular disk are already determined by its values on the boundary of this disc. A strong generalization of this is the residue theorem.
- 2.1 evidence
- 2.2 Example
Cauchy's integral formula for circular disks
Statement
Is open, holomorphic, and a point in a relatively compact disc in, then for all, ie for:
The positively oriented curve is over the edge of.
Evidence
For fixed, the function is defined by all generations. is on continuously and holomorphically on. With the integral theorem of Cauchy is now considered
The function is holomorphic with derivative which vanishes, since the integrand has the master function. Thus is a constant, and because.
Conclusions
For each holomorphic function is: The function value in the center of a circle is the mean value of the function values at the edge of the circle. Use it.
Every holomorphic function is complex differentiable any number of times and each of these derivations is again holomorphic. Expressed with the integral formula means that for and:
Every holomorphic function is locally developable for in a power series.
Using the integral formula for immediately follows that the coefficients are precisely the Taylor coefficients. For the coefficients following estimate is valid when for the following applies:
The set of Liouville (each on all holomorphic bounded function is constant) can very quickly show the integral formula. So you can then easily the fundamental theorem of algebra ( every polynomial splits into linear factors in ) prove.
Evidence
The Cauchy integral formula is partially differentiated, although one must interchange differentiation and integration:
Development of the Cauchy integral formula using the geometric series yields
As for the geometric series converges uniformly, we may integrate term by term, ie Sum and integral reverse. The expansion coefficients are:
For the coefficients of the following estimate holds. There exists a with for; then applies to:
Is on all holomorphic and bounded, ie for all, then applies as before for all:
Since was arbitrary, then for all. Thus it follows from the boundedness of:
This means that every bounded holomorphic function on all is constant ( Liouville's theorem ).
Example
Using the integral formula and integrals can be calculated:
Cauchy's integral formula for polycylinder
The Cauchy integral formula was generalized to the multi-dimensional complex space. Be circular disks in, then a polycylinder is. Be a holomorphic function and then the Cauchy integral formula by
Explained. Since the Cauchy integral theorem does not apply in the multidimensional space, this formula can not be derived analogous to the one-dimensional case from him. This integral formula is therefore derived using induction from the Cauchy integral formula for circular discs. Using the multi- index notation, the formula to back
Be shortened. In multidimensional also applies the formula
For the derivatives of the holomorphic function and the Cauchy inequality
Wherein and the radius of Polyzylinders is. A further generalization of this integral formula is the Bochner - Martinelli formula.
Cauchy's integral formula for cycles
A generalization of the integral formula for circular curves represents the version for cycles is:
If an area is holomorphic and a nullhomologer cycle in, then for all, do not lie on the following integral formula:
The number of turns referred to by.