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 $U\subseteq \mathbb {C}$ be open, $f:U\to \mathbb {C}$ 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 $g:U\to \mathbb {C}$ , such that:

g(z_{o})\not =0\,\wedge \,f(z)=(z-z_{o})^{n}\cdot g(z)

.

Pole of order n

Let $U\subseteq \mathbb {C}$ be open, $f:U\setminus \{z_{o}\}\to \mathbb {C}$ 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 $g:U\to \mathbb {C}$ , such that:

g(z_{o})=w_{o}\in \mathbb {C} \,\wedge \,f(z)=(z-z_{o})^{-n}\cdot g(z){\mbox{ mit }}z\in U\setminus \{z_{o}\}

.

Tasks

Let $U\subseteq \mathbb {C}$ be open, $f:U\to \mathbb {C}$ 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'(z)}{f(z)}}=

Task 2: Zero of order n

Explain why for the term ${\frac {g'(z)}{g(z)}}$ , a neighborhood $D_{\varepsilon }(z_{o})\subset U$ exists where ${\frac {g'}{g}}$ has no singularities.

Task 3: Zero of order n

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

Task 4: Zero of order n

What can you conclude for the following integrals:

\displaystyle \int _{\partial D_{\varepsilon }(z_{o})}{\frac {g'(z)}{g(z)}}\,dz=...

and

\displaystyle \int _{\partial D_{\varepsilon }(z_{o})}{\frac {f'(z)}{f(z)}}\,dz=...

Task 5: Pole of order n

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

\displaystyle \int _{\partial D_{\varepsilon }(z_{o})}{\frac {g'(z)}{g(z)}}\,dz=...

and

\displaystyle \int _{\partial D_{\varepsilon }(z_{o})}{\frac {f'(z)}{f(z)}}\,dz=...

Statement

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

o_{z}(f):=\left\{{\begin{array}{rr}m&z\ {\text{is}}\ {\text{zero order}}\ m{\text{-ter}}\ {\text{Order}}\\-m&z\ {\text{is}}\ {\text{Pole}}\ m{\text{-ter}}\ {\textrm {Order}}\end{array}}\right.

then

{\frac {1}{2\pi i}}\int _{\Gamma }{\frac {f'(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 $g_{z_{0}}\colon U_{z_{0}}\to \mathbb {C}$ such that $g_{z_{0}}(z_{0})\neq 0$ , $U_{z_{0}}\cap (N(f)\cup P(f))={z_{0}}$ , and

f(z)=(z-z_{0})^{o_{z_{0}}(f)}g_{z_{0}}(z)\qquad (\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'(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}}'(z),\qquad z\in U_{z_{0}}

Thus, for $z$ near $z_{0}$ :

{\frac {f'(z)}{f(z)}}={\frac {o_{z_{0}}(f)}{z-z_{0}}}+{\frac {g'{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 $f'/f$ , and

\mathrm {res} _{z_{0}}{\frac {f'}{f}}=o_{z_{0}}(f)

The claim follows by the Residue Theorem.

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Date: 01/07/2024

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