Alphabeta MathHow statement and proof provenance work
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Chordal local uniform convergence and meromorphic normality
Definition
Let be a plane domain and let . The sequence converges chordally locally uniformly to when for every compact set , where is the chordal metric of The chordal metric on the Riemann sphere. For the empty compact set the displayed supremum is defined to be .
A family of meromorphic maps is meromorphically normal when every sequence in has a subsequence converging chordally locally uniformly either to a meromorphic map or to the constant map .
Depends on
Used by
- The family e^(nz) converges chordally to infinity on the right half-plane without being holomorphically normal there Counterexample
- A chordally locally uniform meromorphic limit is meromorphic or identically infinity Theorem
- Local chordal equicontinuity is equivalent to meromorphic normality on compact exhaustions Theorem
Dependency tree · two levels
4 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
- Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
- Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
- Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)