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A contour formula for a locally single-valued holomorphic inverse

Statement

Let $Ω \subseteq C$ be open, let $f$ be holomorphic on $Ω$ , let $Γ$ be a closed complex contour null-homologous in $Ω$ , and let $w \in C$ satisfy $f (z) \neq w$ for every $z \in Γ^{*}$ . Suppose $f (z) = w$ has exactly one solution $a$ inside $Γ$ , that solution is simple, and $n (Γ, a) = 1$ . Then

$a = \frac{1}{2 π i} \int_{Γ} \frac{ζ f^{'} (ζ)}{f (ζ) - w} d ζ .$

Thus, on a contour enclosing exactly one simple preimage branch, the inverse value is recovered by a contour integral.

Facts & Assumptions

Given: A holomorphic function $f$ on an open set $Ω$ , a closed null-homologous contour $Γ$ , and a value $w$ satisfying the hypotheses of the statement.

[L1]

The weighted argument principle multiplies each zero contribution by the value of the holomorphic test function there (The weighted argument principle).

[L2]

The preimage-count corollary identifies the zeros of $f - w$ inside $Γ$ (The argument principle counts preimages of a target value).

Proof

technique · direct

1.1

Because $f$ is holomorphic, the meromorphic function $f - w$ has no poles. The hypotheses say that its only zero inside $Γ$ is the simple zero $a$ .

given

2.1

Apply [L1] to the meromorphic function $f - w$ and the holomorphic test function $g (ζ) = ζ$ . By step 1.1, there is only one zero contribution, its multiplicity is $1$ , and the additional hypothesis $n (Γ, a) = 1$ makes that contribution exactly $a$ . The left-hand side is exactly the displayed contour integral.

step 1.1L1

3.1

Therefore the contour integral equals $a$ . The role of [L2] is to identify the unique enclosed zero as the unique preimage of $w$ .

step 1.1step 2.1L2∎

Depends on

Used by

The inverse contour formula recovers a local inverse value Example

Dependency tree · two levels

7 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

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