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spherical derivative
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(Definition)
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Let $G \subset {\mathbb{C}}$ be a domain.
Definition 1 Let $f \colon G \to {\mathbb{C}}$ be a meromorphic function, then the spherical derivative of $f$ denoted $f^\sharp$ is defined as \begin{equation*} f^\sharp(z) := \frac{2\lvert f'(z) \rvert}{1+\lvert f(z) \rvert^2} \end{equation*}for $z$ where $f(z) \not= \infty$ and when $f(z) = \infty$ define \begin{equation*} f^\sharp(z) = \lim_{\zeta \to z} f^\sharp(\zeta) . \end{equation*}
The second definition makes sense since a meromorphic functions has only isolated poles, and thus $f^\sharp(\zeta)$ is defined by the first equation when we are close to $z$ Some basic properties of the spherical derivative are as follows.
Proposition 1 If $f \colon G \to {\mathbb{C}}$ is a meromorphic function then
Note that sometimes the spherical derivative is also denoted as $\mu(f)(z)$ rather then $f^\sharp(z)$
- 1
- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
- 2
- Theodore B. Gamelin. Complex Analysis. Springer-Verlag, New York, New York, 2001.
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"spherical derivative" is owned by jirka.
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Cross-references: continuous function, properties, equation, poles, isolated, function, meromorphic, domain
There is 1 reference to this entry.
This is version 4 of spherical derivative, born on 2004-04-16, modified 2006-06-19.
Object id is 5773, canonical name is SphericalDerivative.
Accessed 2206 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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