Modular multiplicative inverse: Difference between revisions
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In the [[mathematics|mathematical]] field of [[complex analysis]], the '''Bloch space''', named after [[André Bloch (mathematician)|André Bloch]] and denoted <math>\mathcal{B}</math> or ℬ, is the space of [[holomorphic function]]s ''f'' defined on the [[open set|open]] [[unit disc]] '''D''' in the complex plane, such that the function | |||
: <math>(1-|z|^2)|f^\prime(z)|</math> | |||
is bounded.<ref>{{eom|last=Wiegerinck|first=J.|id=Bloch_function|title=Bloch function}}</ref> <math>\mathcal{B}</math> is a [[Banach space]], with the norm defined by | |||
: <math> \|f\|_\mathcal{B} = |f(0)| + \sup_{z \in \mathbf{D}} (1-|z|^2) |f'(z)|. </math> | |||
This is referred to as the '''Bloch norm''' and the elements of the Bloch space are called '''Bloch functions'''. | |||
==Notes== | |||
{{Reflist}} | |||
{{DEFAULTSORT:Bloch Space}} | |||
[[Category:Complex analysis]] | |||
{{Mathanalysis-stub}} | |||
Revision as of 04:28, 2 February 2014
In the mathematical field of complex analysis, the Bloch space, named after André Bloch and denoted or ℬ, is the space of holomorphic functions f defined on the open unit disc D in the complex plane, such that the function
is bounded.[1] is a Banach space, with the norm defined by
This is referred to as the Bloch norm and the elements of the Bloch space are called Bloch functions.
Notes
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