Complex Analysis/Example Computation with Laurent Series
In this learning resource, rational functions are developed into Laurent series to extract the residue.
From a Rational Function to a Laurent Series
editInitially, a simple rational function of the following form is given:
- with
The goal is to develop it into a Laurent series with the expansion point ..
Definition of Constants
editThe following constants are defined to better illustrate the operations:
Transformation into a Laurent Series
editLet , then:
- :
The residue ,since in the Laurent expansion, the principal part coefficients are all zero (i.e., the principal part vanishes).
Tasks
edit- Why is the condition required for the above calculation Laurent Series (or power series) ?
- Compute the Laurent series for and determine the Residue of the Laurent expansion for in at!***
Factored Powers with Expansion Point in the Denominator
editDefinition of the Function
editFirst,we are given a simple rational function of the form:
- mit
The goal is to develop it into a Laurent series with the expansion point .
Definition of Constants
editThe following constants are defined to better illustrate the operations:
Transformation into a Laurent Series
editthe residue .
Laurent Series with Infinite Principal Part Terms
editA simple rational function of the following form is given:
- with
The goal is to develop it into a Laurent series with the expansion point .
Definition of Constants
editThe following constants are defined for better clarity:
Transformation into a Laurent Series with
editThe residue
Transformation into a Laurent Series with
editThe residue for is erhält man
See Also
edit
Page information
editTranslation and Version Control
editThis page was translated based on the following mit Laurentreihen Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:
- Source: Kurs:Funktionentheorie/Beispielrechnung mit Laurentreihen - URL:https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Integrationsweg
- Date: 11/20/2024