Math 2.0

Functions of a complex variable Definition

Pole

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 a pole of ff if ∣f(z)∣→∞|f(z)| \to \infty as z→cz \to c. This means that for each K>0K > 0 there is δ>0\delta > 0 such that ∣f(z)∣>K|f(z)| > K for each zz with 0<∣z−c∣<δ0 < |z - c| < \delta.

Remarks

The point cc is a pole of ff if and only if f(z)=g(z)(z−c)mf(z) = \frac{g(z)}{(z - c)^m} near cc, for an integer m≥1m \ge 1 and a function gg that is holomorphic near cc with g(c)≠0g(c) \neq 0. The integer mm is the order of the pole. Equivalently, 1/f1/f extends to a holomorphic function near cc with a zero at cc. For example, 1z2(z−1)\frac{1}{z^2 (z - 1)} has a pole of order 22 at 00 and a pole of order 11, a simple pole, at 11.

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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 singularitydefinitionPole, definition, no evidence levelPoledefinition
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  1. Complex numbers object
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  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. Pole definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Zeros_and_poles, Definitions
  2. Web https://en.wikipedia.org/wiki/Zeros_and_poles, lead section
About this page
Tag0WPQYW8MTE
URIm2://0WPQYW8MTE
Aliasessimple-pole
Kinddefinition
WritingW0 stub
MSC30D30
DefinesPole
UsesIsolated singularity, Holomorphic function

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