CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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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.

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Every biholomorphic self-map of the complex plane is affine

Statement

Every biholomorphic self-map of the complex plane is affine: if $f : C \to C$ is biholomorphic, then there are $a, b \in C$ with $a \neq 0$ and $f (z) = a z + b .$

Facts & Assumptions

Given: A biholomorphic map $f : C \to C$ .

[L1]

A meromorphic self-map of the sphere is rational, and a bijective rational sphere map has degree $1$ (Meromorphic functions on the Riemann sphere are exactly the rational functions, A nonconstant rational map has total fibre multiplicity equal to its degree).

[L2]

In the one-point compactification, continuity at $\infty$ is exactly preservation of compact subsets under inverse images (The one-point (Alexandroff) compactification $X^{*} = X \cup {\infty}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ , $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff).

Proof

technique · direct

1.1

Since $f$ and $f^{- 1}$ are homeomorphisms of $C$ , [L2] extends $f$ to a sphere homeomorphism $F$ with $F (\infty) = \infty$ . In the infinity chart, the reciprocal expression $1 / f (1 / w)$ is bounded near $0$ , so the removable-singularity theorem makes $F$ meromorphic at $\infty$ . Thus $F$ is a meromorphic self-map of the sphere.

L2given

2.1

Fact [L1] makes $F$ a rational sphere map of degree $1$ , hence Möbius. Because $F (\infty) = \infty$ , its denominator has zero $z$ -coefficient, so restricting back to $C$ gives $f (z) = a z + b$ with $a \neq 0$ .

L1givenalgebra∎

Depends on

Used by

Dependency tree · two levels

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Sources

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)