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The argument principle for an admissible null-homologous cycle

Statement

Let $Ω \subseteq C$ be open, let $f$ be meromorphic on $Ω$ , and let $Γ$ be admissible for the residue theorem in $Ω$ . Suppose in addition that $f$ is not identically zero on any connected component of $Ω$ and that $f (z) \neq 0$ for every $z \in Γ^{*}$ . Then

$\frac{1}{2 π i} \int_{Γ} \frac{f^{'} (z)}{f (z)} d z = Z (f, Γ) - P (f, Γ),$

where the weighted zero and pole counts are those of Zero and pole counts weighted by multiplicity and winding number.

Only finitely many terms in those weighted counts are nonzero.

As in Zero and pole counts weighted by multiplicity and winding number, meromorphicity on the possibly disconnected open set $Ω$ is understood componentwise.

Facts & Assumptions

Given: An open set $Ω$ , a meromorphic function $f$ on $Ω$ that is not identically zero on any connected component, and an admissible cycle $Γ$ in $Ω$ such that $f$ has no zero on $Γ^{*}$ .

[L1]

Away from the zeros and poles of $f$ , the logarithmic derivative $f^{'} / f$ is holomorphic (The logarithmic derivative of a meromorphic function).

[L2]

At a zero of order $m$ , the logarithmic derivative has residue $m$ , and at a pole of order $m$ it has residue $- m$ (The logarithmic derivative has residue equal to local order).

[L3]

A meromorphic function admissible for a cycle has only finitely many poles with nonzero index (Only finitely many singularities contribute to the residue sum of an admissible cycle).

[L4]

The residue theorem for an admissible null-homologous cycle reads $\int_{Γ} g (z) d z = 2 π i \sum_{c} n (Γ, c) Res (g, c)$ with only finitely many nonzero terms (The residue theorem for a null-homologous cycle).

Proof

technique · direct

1.1givenL1L2

By [L1], the function $g : = f^{'} / f$ is holomorphic away from the zeros and poles of $f$ . By [L2], every zero or pole of $f$ becomes a simple pole of $g$ . Because $f$ has neither zeros nor poles on $Γ^{*}$ , the cycle $Γ$ is admissible for $g$ as well.

2.1step 1.1L3

Applying [L3] to $g$ shows that only finitely many zeros or poles of $f$ have nonzero index with respect to $Γ$ . Therefore the sums defining $Z (f, Γ)$ and $P (f, Γ)$ are finite.

3.1step 1.1step 2.1L2L4∎

By [L4] applied to $g$ , $\frac{1}{2 π i} \int_{Γ} \frac{f^{'} (z)}{f (z)} d z = \sum_{c} n (Γ, c) Res ⁣ (\frac{f^{'}}{f}, c),$ where $c$ ranges over the poles of $g$ . Splitting those poles into zeros and poles of $f$ and then using [L2] turns the right-hand side into $Z (f, Γ) - P (f, Γ)$ .

The argument principle for an admissible null-homologous cycle

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources