ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The logarithm surface admits the standard helicoid model

Example

Write $w = x + i y$ . Under the biholomorphism $Λ : R_{\log} \to C$ of The Riemann surface of the logarithm is the complex plane over the punctured plane via exp, the logarithm surface is modeled in $R^{3}$ by the helicoid

$H (w) : = (e^{x} \cos y, e^{x} \sin y, y) .$

Its projection to the first two coordinates is the punctured-plane point $e^{w} \in C^{\times}$ .

Facts & Assumptions

Given: The biholomorphism $Λ : R_{\log} \to C$ .

[L1]

The logarithm surface is biholomorphic to $C$ , and its projection corresponds to $\exp$ (The Riemann surface of the logarithm is the complex plane over the punctured plane via exp).

Verification

technique · direct

1.1

Writing $w = x + i y$ , Euler's formula gives $e^{w} = e^{x} (\cos y + i \sin y)$ . So the first two coordinates of $H (w)$ are exactly the complex number $e^{w}$ .

givenalgebra

2.1

By [L1], the point of the logarithm surface corresponding to $w$ projects to $e^{w}$ . Step 1.1 therefore identifies the abstract surface with the standard helicoid model in $R^{3}$ , whose vertical coordinate records the argument before taking it modulo $2 π$ .

L1step 1.1∎

Depends on

The Riemann surface of the logarithm is the complex plane over the punctured plane via exp

Used by

FALSE: the Riemann surface of a multivalued function is automatically a subset of C squared False statement

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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §4.3 (standard reference, not scraped)