ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Near $(0, 1)$ , the equation $z_{1}^{2} + z_{2}^{2} = 1$ is a holomorphic graph

Example

For

$f (z_{1}, z_{2}) = z_{1}^{2} + z_{2}^{2} - 1,$

the equation $f (z_{1}, z_{2}) = 0$ defines $z_{2}$ as a holomorphic function of $z_{1}$ near $(0, 1)$ .

Facts & Assumptions

Given: The holomorphic function $f (z_{1}, z_{2}) = z_{1}^{2} + z_{2}^{2} - 1$ and the point $(0, 1)$ .

[L1]

The holomorphic implicit function theorem applies when the derivative in the dependent variable is invertible (The holomorphic implicit function theorem).

Verification

technique · direct

1.1

One has $f (0, 1) = 0$ and $\frac{\partial f}{\partial z_{2}} (0, 1) = 2 \neq 0.$

givenalgebra

2.1

Therefore [L1] yields neighbourhoods of $0$ and $1$ and a unique holomorphic function $φ$ such that $f (z_{1}, φ (z_{1})) = 0$ for all nearby $z_{1}$ . So near $(0, 1)$ the zero set is the holomorphic graph $z_{2} = φ (z_{1})$ .

step 1.1L1∎

Depends on

The holomorphic implicit function theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Jaap Korevaar and Jan Wiegerinck, Several Complex Variables, Section 4.2 (standard reference, not scraped)

Near (0,1)(0,1), the equation z12+z22=1z12​+z22​=1 is a holomorphic graph