Alphabeta MathHow statement and proof provenance work
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.
Meromorphic functions on the Riemann sphere are exactly the rational functions
Statement
A map on the Riemann sphere is meromorphic if and only if it is rational. Precisely, is meromorphic on the Riemann sphere exactly when it is not identically and there are polynomials , not both zero and chosen coprime, such that on the finite chart
Facts & Assumptions
Given: A sphere-valued map on .
A pole is exactly a finite nonzero principal part in the Laurent expansion, and the same criterion applies at in the -chart (Characterizations of poles).
Every bounded entire function is constant (Liouville's theorem: every bounded entire function is constant).
The poles of a meromorphic plane function form a closed discrete set (Poles of a meromorphic function form a closed discrete set and are at most countable).
Proof
If is rational with coprime polynomials, then it is holomorphic on away from the zeros of , those zeros are poles of finite order, and in the infinity chart the expression is meromorphic at . So every rational map is meromorphic on the sphere.
Conversely, assume is sphere-meromorphic. Meromorphy at gives a radius beyond which there are no finite poles, and [L3] makes the remaining finite pole set discrete. Covering that pole set by isolating discs and using compactness of the containing closed disc shows that only finitely many finite poles occur.
Fact [L1] gives a finite principal part at each finite pole and a finite principal part in the infinity chart. Subtracting all of those principal parts leaves an entire function that is bounded near , hence bounded on all of ; [L2] therefore makes the remainder constant. So is a rational function.
The first step proves the rational-to-meromorphic direction and the latter two steps prove the converse, so sphere-meromorphic functions are exactly rational functions.
Depends on
- Meromorphic functions on the Riemann sphere
- Liouville's theorem: every bounded entire function is constant
- $X^{*}$ 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
- Poles of a meromorphic function form a closed discrete set and are at most countable
- Characterizations of poles
Used by
- Every biholomorphic self-map of the complex plane is affine Corollary
- The exponential function is meromorphic on C but not meromorphic on the Riemann sphere Counterexample
- The degree of a rational self-map of the Riemann sphere Definition
- Every biholomorphic self-map of the punctured plane is of the form az or a/z Theorem
- Every biholomorphic self-map of the Riemann sphere is Möbius Theorem
Dependency tree · two levels
39 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 §§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)