Complex Analysis/Residue Theorem
The residue theorem states, how to calculate the integral of a holomorphic function using its Residuals .
Statement
[edit | edit source]Let be a holomorphic function in a region except for a discrete set of isolated singularities , and let be a null-homologous Chain in that does not intersect any point of . Then, the following holds:
Proof
[edit | edit source]The sum in the statement of the residue theorem is finite because can enclose only finitely many points of the discrete set of singularities.
Step 1 - Reduction to a finite number of summands
[edit | edit source]Let be the points in for which . The singularities in that are not enclosed are denoted by .
Step 2 - Nullhomologous cycle
[edit | edit source]is assumed to be null-homologous in . By the definition of ,is also null-homologous in .
Step 3 - Principal parts of the Laurent series
[edit | edit source]For the singularities with and , let
be the main part of the Laurent Expansion of around . The function is holomorphic on .
Step 4 - Subtraction of principal parts
[edit | edit source]Subtracting all the principal parts corresponding to from the given function , we obtain
a function on that now has only removable singularities.
Step 5 - Holomorphic extension to
[edit | edit source]If the singularities are Isolated singularity on , can be extended holomorphically to all .
Step 6 - Application of Cauchy's integral theorem
[edit | edit source]By the Cauchy Integral Theorem for , we have
so, by the definition of ,
Step 7 - Calculation of integrals of the principal parts
[edit | edit source]The computation of the integral over reduces to computing the integrals of the principal parts for . Using the linearity of the integral, we have:
the terms For , have antiderivatives, so .
Step 8 - Calculation of integrals of the principal parts
[edit | edit source]Finally, the computation of the integrals of the principal parts yields, using the definition of the Winding number:
after.
Step 9 - Calculation of the integrals of the residues
[edit | edit source]Thus, the statement follows as:
Questions about the residue theorem
[edit | edit source]- Let be a meromorphic function (i.e., holomorphic except for a discrete set of singularities in ). Why does the cycle enclose only finitely many poles?
Applications
[edit | edit source]The Zeros and poles counting integral counts the zeros and poles of a meromorphic function.
See also
[edit | edit source]Page Information
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Translation and Version Control
[edit | edit source]This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:
- Source: Kurs:Funktionentheorie/Residuensatz - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuensatz
- Date: 01/05/2024