Lagmental Vicfred

Rouché's Theorem Preserves the Number of Zeros by Vicfred

Last updated: Mon 10 September 2012

tags: mathematics, complex analysis, contour integration, residues

A strict boundary domination lets a holomorphic perturbation keep the same interior zero count. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

The data

A holomorphic function \(f:U\to\mathbf C\) has a complex derivative independent of direction. Cauchy's integral formula controls each value \(f^{(n)}(a)\) from boundary data.

$$ |g(z)|<|f(z)|\qquad(z\in\gamma) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ f\ \text{and }f+g\ \text{have the same number of zeros inside }\gamma $$

Derivation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ |10z+1|\le21<32=|z^5|\quad\text{on }|z|=2\Longrightarrow z^5+10z+1\text{ has five zeros in }|z|<2 $$

Invariant content

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad |g(z)|<|f(z)|\qquad(z\in\gamma),\\[5pt] \mathsf{C}\;&:\quad f\ \text{and }f+g\ \text{have the same number of zeros inside }\gamma. \end{aligned} $$

Scope

Contour formulas require orientation, winding number, and hypotheses about singularities. A pole on the contour cannot be ignored.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f\ \text{and }f+g\ \text{have the same number of zeros inside }\gamma \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Tue 17 April 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.