Math 2.0

Functions of a complex variable Definition

Removable singularity

Builds on Isolated singularityHolomorphic function

Definition

Let cc be an isolated singularity of ff, so that ff is holomorphic on 0<∣z−c∣<r0 < |z - c| < r for some r>0r > 0. The singularity cc is removable if there is a function gg that is holomorphic on the disc ∣z−c∣<r|z - c| < r with g(z)=f(z)g(z) = f(z) for each zz with 0<∣z−c∣<r0 < |z - c| < r.

Remarks

The standard example is f(z)=sin⁡zzf(z) = \frac{\sin z}{z} at 00. With g(0)=1g(0) = 1, the function gg is the sum of 1−z23!+z45!−⋯1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \cdots, which is holomorphic. The singularity is removable if and only if the principal part of ff at cc is 00. Riemann's theorem says that the singularity is removable if ff is bounded near cc.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionIsolated singularity, definition, no evidence levelIsolated singularitydefinitionRemovable singularity, definition, no evidence levelRemovable singularitydefinition
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  1. Complex numbers object
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  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Modulus of a complex number definition
  10. Complex differentiability definition
  11. Holomorphic function definition
  12. Isolated singularity definition
  13. Removable singularity definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Removable_singularity, lead section
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Tag0P18JC72RE
URIm2://0P18JC72RE
Kinddefinition
WritingW1 stated
MSC30D30
DefinesRemovable singularity
UsesIsolated singularity, Holomorphic function

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