Riemann's Removable Singularity Theorem states that if a function is holomorphic on a punctured neighborhood of a point and remains bounded as that point is approached, the singularity is removable, meaning the function extends to a holomorphic function on the whole neighborhood including that point. Named for Bernhard Riemann, it is one of the basic classification tools of complex analysis, distinguishing a removable singularity from the more severe poles and essential singularities a function can otherwise have.
Facts
StatementFor f holomorphic on D minus a point a, extending holomorphically over a, extending continuously over a, being bounded near a, and the limit of (z - a)f(z) being 0 as z approaches a are equivalent. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Wolfram MathWorld: Riemann's Removable Singularity Theorem -- from Wolfram MathWorld
Named After
Proved By
Sources
1. Removable singularity (Wikipedia)
Riemann's theoremQuote, Riemann's theorem
f is holomorphically extendable over a.
Wolfram MathWorld: Riemann's Removable Singularity Theorem -- from Wolfram MathWorld
Wolfram MathWorldIn Branch: Complex Analysis, Subject classificationsQuote, In Branch: Complex Analysis, Subject classifications
Calculus and Analysis Complex Analysis
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