Isolated singularity

Isolated singularities are considered in the mathematical branch of function theory. Isolated singularities are special isolated points in the source set of a holomorphic function. A distinction is made between isolated singularities elevating singularities, poles and essential singularities.

Definition

It should be an open subset. It should also be a holomorphic complex-valued function.

Then called isolated singularity of.

Each isolated singularity belongs to one of the following three classes:

• The point is called removable singularity if it is to be continued holomorphically. After the Riemannian Hebbarkeitssatz this is for example the case when in an environment of limited.
• The point is, pole or pole if no removable singularity and there is a natural number, so has a removable singularity at. If the minimum is selected, it is said, was in a pole of order.
• Otherwise called an essential singularity of.

Elevating singularities and poles are also summarized under the term except essential singularity.

Isolated singularities and Laurent series

The type of singularity is also reflected in the Laurent series

Read of in:

• A removable singularity exists if and only if the principal part vanishes, ie for all negative integers.
• A pole of order exists if and only if the main part breaks off after members, ie for all.
• An essential singularity exists if and only if infinitely many terms with negative exponents do not disappear.

Statements about the properties of holomorphic functions of essential singularities make the Great set of Picard and a simpler special case of the set of Casorati - Weierstrass.

Examples

It should be and

• Can be continued by steadily on, so it has a removable singularity at.
• Has in a first-order pole, because it can be continued by steadily on.
• Has an essential singularity at, because of, for fixed always be unlimited or because not disappear in the Laurent series for infinitely many terms of the main part, because it is

Swell

• Eberhard Freitag & Rolf Busam: Function Theory 1, Springer- Verlag, Berlin, ISBN 3-540-67641-4.

• Function theory