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Marty's theorem
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(Theorem)
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Here normal convergence (convergence on compact subsets) is given using the spherical metric and not the standard metric of the complex plane. That is, if $\sigma$ is the spherical metric then we will say $f_n \to f$ normally if $\sigma(f_n(z),f(z))$ converges to 0 uniformly on compact subsets.
A related theorem can be stated.
Theorem 2 If $f_n(z) \to f(z)$ uniformly in the spherical metric on compact subsets of $G \subset {\mathbb{C}}$ then $f_n^\sharp(z) \to f^\sharp(z)$ uniformly on compact subsets of $G$ .
Here $f^\sharp$ denotes the spherical derivative of $f$ .
- 1
- Theodore B. Gamelin. Complex Analysis. Springer-Verlag, New York, New York, 2001.
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"Marty's theorem" is owned by jirka.
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Cross-references: theorem, converges, complex plane, standard metric, spherical metric, normal convergence, compact subset, bounded, spherical derivatives, domain, normal family, functions, meromorphic
There is 1 reference to this entry.
This is version 4 of Marty's theorem, born on 2004-04-16, modified 2006-06-18.
Object id is 5774, canonical name is MartysTheorem.
Accessed 2299 times total.
Classification:
| AMS MSC: | 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory) |
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Pending Errata and Addenda
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