Math 2.0

Functions of a complex variable Singularities and residues Theorem

Riemann's theorem on removable singularities

Builds on Isolated singularityRemovable singularityHolomorphic function

Statement

Let c∈ℂc \in \mathbb{C}, let r>0r > 0, and let ff be holomorphic on the punctured disc 0<∣z−c∣<r0 < |z - c| < r. Assume that ff is bounded near cc: there are M≥0M \ge 0 and δ∈(0,r)\delta \in (0, r) with ∣f(z)∣≤M|f(z)| \le M for 0<∣z−c∣<δ0 < |z - c| < \delta. Then cc is a removable singularity of ff. Moreover, the limit lim⁡z→cf(z)\lim_{z \to c} f(z) exists, and the extension of ff by this value at cc is holomorphic on ∣z−c∣<r|z - c| < r.

Remarks

The following are equivalent: cc is removable; ff has a continuous extension to cc; ff is bounded near cc; lim⁡z→c(z−c)f(z)=0\lim_{z \to c} (z - c) f(z) = 0. Proof idea: h(z)=(z−c)2f(z)h(z) = (z - c)^2 f(z), with h(c)=0h(c) = 0, is complex differentiable at cc with h′(c)=0h'(c) = 0. So h(z)=∑n≥2an(z−c)nh(z) = \sum_{n \ge 2} a_n (z - c)^n, and f(z)=∑n≥2an(z−c)n−2f(z) = \sum_{n \ge 2} a_n (z - c)^{n-2}. For example, sin⁡zz\frac{\sin z}{z} is bounded near 00, so 00 is removable.

Reading path

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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 singularitydefinitionRiemann's theorem on removable singularities, statement, no evidence levelRiemann's theorem on …statement
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
  5. Supremum definition
  6. Completeness axiom statement
  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
  14. Riemann's theorem on removable singularities statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Removable_singularity, Riemann's theorem
About this page
Tag0BSQDZWF34
URIm2://0BSQDZWF34
Aliasesriemann-removable-singularity-theorem
Kindstatement, theorem
WritingW0 stub
MSC30D30
StatusThis copy of the site has no build data. Run m2 status 0BSQDZWF34.
UsesIsolated singularity, Removable singularity, Holomorphic function

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