Complex Analysis/Zero and Pole counting integral

The integral counting zeros and poles counts, as the name suggests, the zeros and poles of a meromorphic function along with their multiplicities. More precisely:

Zero of order n

Let $U\subseteq ℂ$ be open, $f:U\to ℂ$ a holomorphic function, and $z_o\in U$. The function $f$ has $z_o$ a zero of order $n\in \mathbb{N}$ at $z_o$ if there exists a holomorphic function $g:U\to ℂ$, such that:

$g(z_o)≠0\wedge f(z)=(z-z_o{)}^n\cdot g(z)$.

Pole of order n

Let $U\subseteq ℂ$ be open, $f:U\setminus \{z_o\}\to ℂ$ a holomorphic function, and $z_o\in U$. The function $f$ has $z_o$ a pole of order $n\in \mathbb{N}$ at $z_o$ if there exists a holomorphic function $g:U\to ℂ$, such that:

$g(z_o)=w_o\in ℂ\wedge f(z)=(z-z_o{)}^{-n}\cdot g(z)\text{ mit }z\in U\setminus \{z_o\}$.

Tasks

Let $U\subseteq ℂ$ be open, $f:U\to ℂ$ a holomorphic function, and $z_o\in U$. Furthermore, let $f$ have $z_o$ a zero of order $n\in \mathbb{N}$ at $z_o$.

Task 1: Zero of order n

Using the definition of the order of a zero, compute the expression for $z\in U$:

$\frac{f^{\prime }(z)}{f(z)}=$

Task 2: Zero of order n

Explain why for the term $\frac{g^{\prime }(z)}{g(z)}$, a neighborhood $D_{\epsilon }(z_o)\subset U$ exists where $\frac{g^{\prime }}{g}$ has no singularities.

Task 3: Zero of order n

Explain why $\frac{g^{\prime }(z)}{g(z)}$ does not necessarily need to be defined on the entire set $U\subseteq ℂ$.

Task 4: Zero of order n

What can you conclude for the following integrals:

${\int }_{\partial D_{\epsilon }(z_o)}\frac{g^{\prime }(z)}{g(z)}dz=\mathrm{...}$

and

${\int }_{\partial D_{\epsilon }(z_o)}\frac{f^{\prime }(z)}{f(z)}dz=\mathrm{...}$

Task 5: Pole of order n

Apply the calculations and explanations to poles of order $n\in \mathbb{N}$ and compute the integrals:

${\int }_{\partial D_{\epsilon }(z_o)}\frac{g^{\prime }(z)}{g(z)}dz=\mathrm{...}$

and

${\int }_{\partial D_{\epsilon }(z_o)}\frac{f^{\prime }(z)}{f(z)}dz=\mathrm{...}$

Statement

Let $U\subseteq ℂ$ be open, and $f\in ℳ(U)$. Let $N(f)$ be the set of zeros and $P(f)$ the set of poles of $f$. Let $\mathrm{\Gamma }\in C(U)$ be a Chain that encircles each zero and each pole of $f$ exactly once in the positive orientation Winding number , i.e., $n(\mathrm{\Gamma },z)=1$ for each $z\in N(f)\cup P(f)$. For $z\in N(f)\cup P(f)$, we set:

$o_z(f):=\{\begin{array}{cc}m& z\text{is}\text{zero order}m\text{-ter}\text{Order}\\ -m& z\text{is}\text{Pole}m\text{-ter}\text{Order}\end{array}$

then

$\frac{1}{2\pi i}{\int }_{\mathrm{\Gamma }}\frac{f^{\prime }(z)}{f(z)}dz=\sum _{z\in N(f)\cup P(f)}{o}_z(f).$

Proof

For each $z_0\in N(f)\cup P(f)$, there exists a neighborhood $U_{z_0}$ and a holomorphic function $g_{z_0}:U_{z_0}\to ℂ$ such that $g_{z_0}(z_0)\ne 0$, $U_{z_0}\cap (N(f)\cup P(f))=z_0$, and

$f(z)=(z-z_0{)}^{o_{z_0}(f)}{g}_{z_0}(z)(\star )$

holds.

Proof 1: Holomorphicity and Application of Residue Theorem

The integrand is holomorphic everywhere in $U$, except possibly at $N(f)\cup P(f)$. By the Residue Theorem, it suffices to compute the residues of $f$ at the points of $N(f)\cup P(f)$.

Proof 2: Residue for Zeros/Poles

Let $z_0\in N(f)\cup P(f)$. Differentiating $(\star )$, we obtain:

$f^{\prime }(z)=o_{z_0}(f)(z-z_0{)}^{o_{z_0}(f)-1}{g}_{z_0}(z)+(z-z_0{)}^{o_{z_0}(f)}{g}_{{z_0}^{\prime }}(z),z\in U_{z_0}$

Thus, for $z$ near $z_0$:

$\frac{f^{\prime }(z)}{f(z)}=\frac{o_{z_0}(f)}{z-z_0}+\frac{g^{\prime }{z}_0(z)}{g{z}_0(z)},z\in U_{z_0}$with

Proof 3: Application of Residue Theorem

The second term is holomorphic, so $z_0$ is a simple pole of $f^{\prime }/f$, and

${\mathrm{r}\mathrm{e}\mathrm{s}}_{z_0}\frac{f^{\prime }}{f}=o_{z_0}(f)$

The claim follows by the Residue Theorem.

Page Information

You can display this page as Wiki2Reveal slides

Wiki2Reveal

The Wiki2Reveal slides were created for the Complex Analysis' and the Link for the Wiki2Reveal Slides was created with the link generator.

• This page is designed as a PanDocElectron-SLIDE document type.
• Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Zero%20and%20Pole%20counting%20integral
• see Wiki2Reveal for the functionality of Wiki2Reveal.

Translation and Version Control

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/Null- und Polstellen zählendes Integral - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Null-_und_Polstellen_zählendes_Integral

• Date: 01/07/2024