Math 2.0

Functions of a complex variable Definition

Order of a pole

Builds on PoleOrder of a zero

Definition

Let cc be a pole of ff. The order of the pole cc is the unique integer m≥1m \ge 1 with the following property. On some punctured disc 0<∣z−c∣<δ0 < |z - c| < \delta, f(z)=g(z)(z−c)m,f(z) = \frac{g(z)}{(z - c)^m}, where gg is holomorphic on the disc ∣z−c∣<δ|z - c| < \delta and g(c)≠0g(c) \neq 0. A pole of order 11 is a simple pole.

Remarks

The order of the pole of ff at cc is the order of the zero of 1/f1/f at cc. In the Laurent series ∑an(z−c)n\sum a_n (z - c)^n of ff at cc, the order is mm if and only if a−m≠0a_{-m} \neq 0 and an=0a_n = 0 for each n<−mn < -m. Many authors give a pole of order mm the signed order −m-m. Then zeros have positive order and poles have negative order. For example, 1sin⁡2z\frac{1}{\sin^2 z} has a pole of order 22 at each point kπk\pi with k∈ℤk \in \mathbb{Z}.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness 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 functiondefinitionCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementIsolated singularity, definition, no evidence levelIsolated singularitydefinitionZero of a holomorphic function, definition, no evidence levelZero of a holomorphic…definitionPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statementPole, definition, no evidence levelPoledefinitionHolomorphic functions are analytic, statement, no evidence levelHolomorphic functions…statementHolomorphic functions are infinitely differentiable, statement, no evidence levelHolomorphic functions…statementOrder of a zero, definition, no evidence levelOrder of a zerodefinitionOrder of a pole, definition, no evidence levelOrder of a poledefinition
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Complex exponential definition
  5. Contour integral definition
  6. Power series in the complex plane definition
  7. Real and imaginary parts definition
  8. Supremum definition
  9. Circle integral definition
  10. Disc of convergence definition
  11. Completeness axiom statement
  12. Existence of square roots of nonnegative reals statement
  13. Square root definition
  14. Modulus of a complex number definition
  15. Complex differentiability definition
  16. Holomorphic function definition
  17. Cauchy's integral formula statement
  18. Isolated singularity definition
  19. Zero of a holomorphic function definition
  20. Power series are holomorphic in the disc of convergence statement
  21. Pole definition
  22. Holomorphic functions are analytic statement
  23. Holomorphic functions are infinitely differentiable statement
  24. Order of a zero definition
  25. Order of a pole definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Zeros_and_poles, Definitions
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