CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A real-valued holomorphic function on a domain is constant

Statement

If $f : U \to C$ is holomorphic on a domain $U$ and $f (U) \subseteq R$ , then $f$ is constant.

Facts & Assumptions

Given: A domain $U$ and a holomorphic $f = u + i v : U \to C$ with $v = 0$ .

[L1]

A holomorphic map satisfies $u_{x} = v_{y}$ , $u_{y} = - v_{x}$ , and $f^{'} = u_{x} + i v_{x}$ (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[L2]

A holomorphic function with zero derivative on a domain is constant (A holomorphic function with zero derivative on a domain is constant).

Proof

technique · direct

1.1

Since $v = 0$ , both $v_{x}$ and $v_{y}$ vanish. The Cauchy–Riemann equations [L1] then give $u_{x} = u_{y} = 0$ , so $f^{'} = 0$ throughout $U$ .

givenL1algebra

2.1

Apply [L2] to conclude that $f$ is constant on $U$ .

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

17 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, Exercise 2.1.5 (standard reference, not scraped)