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Complex Analysis/Residue Theorem

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The residue theorem states, how to calculate the integral of a holomorphic function using its Residuals .

Statement

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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

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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

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Let be the points in for which . The singularities in that are not enclosed are denoted by .

Step 2 - Nullhomologous cycle

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is assumed to be null-homologous in . By the definition of ,is also null-homologous in .

Step 3 - Principal parts of the Laurent series

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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

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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

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If the singularities are Isolated singularity on , can be extended holomorphically to all .

Step 6 - Application of Cauchy's integral theorem

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By the Cauchy Integral Theorem for , we have

so, by the definition of ,

Step 7 - Calculation of integrals of the principal parts

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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

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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

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Thus, the statement follows as:

Questions about the residue theorem

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  • 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

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The Zeros and poles counting integral counts the zeros and poles of a meromorphic function.

See also

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Page Information

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Translation and Version Control

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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:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuensatz

  • Date: 01/05/2024