Math 2.0

Functions of a complex variable Definition

Isolated singularity

Builds on Holomorphic function

Definition

Let c∈ℂc \in \mathbb{C} and let ff be a complex function. The point cc is an isolated singularity of ff if there is r>0r > 0 such that ff is defined and holomorphic on the punctured disc {z:0<∣z−c∣<r}\{z : 0 < |z - c| < r\}. The value of ff at cc, if there is one, plays no role.

Remarks

Each isolated singularity is of exactly one of three types: removable, a pole, or essential. For example, 00 is an isolated singularity of 1/z1/z, of sin⁡zz\frac{\sin z}{z} and of e1/ze^{1/z}. The point 00 is not an isolated singularity of 1sin⁡(1/z)\frac{1}{\sin(1/z)}, because the poles 1kπ\frac{1}{k\pi} accumulate at 00. It is not an isolated singularity of Log⁡z\operatorname{Log} z either, because Log⁡\operatorname{Log} is not holomorphic on any punctured disc around 00.

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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 singularitydefinition
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  1. Complex numbers object
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  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

Where this leads

42 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Isolated_singularity, lead section
  2. Web https://en.wikipedia.org/wiki/Isolated_singularity, Examples
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