Rouché's theorem

The set of Rouché ( after Eugène Rouché ) is a set of function theory.

He makes a statement about the functions with which one can disturb a holomorphic function, without changing the number of zeros. The version for meromorphic functions makes a similar statement for the difference of zeroes and poles.

The set of Rouché for holomorphic functions

Be two holomorphic functions in the field. In addition, the circular disk, including its edge is included and applies to all points of the edge:

Then the functions and the same number of zeros have ( according to the multiplicities counted ) on.

Note: denotes the open disc with center and radius r.

Symmetric version

Under weakening of the conditions is that two holomorphic functions have the same number of zeros within a limited area with a continuous edge when on the edge of the strict triangle inequality

Applies. Theodor Estermann showed this general formulation for the first time in his book Complex Numbers and Functions.

Application: barriers for Polynomnullstellen

It is a complex polynomial coefficient. The region G is the entire complex plane. It is an index for which the inequality

Is satisfied for at least one. Then perform the functions and the conditions of the set of Rouché for the circle B (0, r). f is different from zero and therefore has exactly one zero of multiplicity k at the origin. It follows that p = f g exactly k zeros ( counted with multiplicity) in the circle B (0, r) possesses.

The set of meromorphic functions for Rouché

Be two in the field of meromorphic functions. Further argues, and that there are no zeros or poles on the edge have; and is for everyone:

Then vote for and the differences

(corresponding to the multiplicity or Polordnung counted ) agree on.

Proof of meromorphic functions

Define.

By assumption:

Since the circle is compact, there is even an open neighborhood of these, so that the inequality is satisfied on U. The break f / g takes on its U values ​​within the unit circle B (0.1 ), therefore also applies:

The open disc contained in the domain of the main branch of the holomorphic logarithm, and:

Now we consider the following integral:

The integrand has a primitive function, so the following applies:

According to the argument principle also applies in extension but the residue theorem:

Wherein the number of zeros of, and the number of poles of call on.

From this the assertion follows: