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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

1.1

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.

givenL1L2

2.1

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.

step 1.1L3

3.1

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 $Z (f, Γ) - P (f, Γ)$ .

step 1.1step 2.1L2L4∎

Depends on

Used by

The argument principle counts preimages of a target value Corollary
The function e^(1/z) shows that essential singularities lie outside the argument principle Counterexample
FALSE: the argument principle ignores multiplicity False statement
The argument-principle integral is the winding number of the image cycle Theorem
The weighted argument principle Theorem

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. W. Howell and J. H. Mathews, Complex Analysis, §8.7, Theorem 8.7.2 (standard reference, not scraped)
J. Lebl, Guide to Cultivating Complex Analysis, §5.4, Theorem 5.4.1 (standard reference, not scraped)