Math 2.0

Functions of a complex variable Definition

Order of a zero or pole

Builds on Meromorphic functionOrder of a zeroOrder of a poleLaurent seriesHolomorphic function

Definition

Let U⊆ℂU \subseteq \C be an open set, let ff be a meromorphic function on UU, and let a∈Ua \in U. Assume that ff is not identically 00 on any punctured disc around aa. The order of ff at aa, written ord⁡af\operatorname{ord}_a f, is the unique integer nn with the following property. There are r>0r > 0 and a holomorphic function gg on {z:∣z−a∣<r}\set{z : \abs{z - a} < r} with g(a)≠0g(a) \neq 0 and f(z)=(z−a)ng(z)f(z) = (z - a)^n g(z) for 0<∣z−a∣<r0 < \abs{z - a} < r. If ff is identically 00 near aa, one sets ord⁡af=∞\operatorname{ord}_a f = \infty.

Remarks

If n>0n > 0, then aa is a zero of order nn; if n<0n < 0, then aa is a pole of order −n-n; if n=0n = 0, then f(a)f(a) is defined and nonzero. Equivalently, nn is the least index with a nonzero coefficient in the Laurent series of ff at aa. Example: f(z)=sin⁡zz3f(z) = \dfrac{\sin z}{z^3} has ord⁡0f=−2\operatorname{ord}_0 f = -2, since sin⁡z=z h(z)\sin z = z \, h(z) with hh holomorphic and h(0)=1h(0) = 1. Non-example: e1/ze^{1/z} is not meromorphic near 00 and has no order there. Its Laurent series at 00 has infinitely many nonzero terms of negative index. The order satisfies ord⁡a(fg)=ord⁡af+ord⁡ag\operatorname{ord}_a(fg) = \operatorname{ord}_a f + \operatorname{ord}_a g and ord⁡a(f+g)≥min⁡(ord⁡af,ord⁡ag)\operatorname{ord}_a(f + g) \ge \min(\operatorname{ord}_a f, \operatorname{ord}_a g). Some authors call it the valuation of ff at aa.

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 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 convergencedefinitionLaurent series, definition, no evidence levelLaurent seriesdefinitionCompleteness 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…statementMeromorphic function, definition, no evidence levelMeromorphic functiondefinitionHolomorphic 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 poledefinitionOrder of a zero or pole, definition, no evidence levelOrder of a zero or po…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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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. Laurent series definition
  12. Completeness axiom statement
  13. Existence of square roots of nonnegative reals statement
  14. Square root definition
  15. Modulus of a complex number definition
  16. Complex differentiability definition
  17. Holomorphic function definition
  18. Cauchy's integral formula statement
  19. Isolated singularity definition
  20. Zero of a holomorphic function definition
  21. Power series are holomorphic in the disc of convergence statement
  22. Pole definition
  23. Holomorphic functions are analytic statement
  24. Meromorphic function definition
  25. Holomorphic functions are infinitely differentiable statement
  26. Order of a zero definition
  27. Order of a pole definition
  28. Order of a zero or pole definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Zeros_and_poles
About this page
TagK2S5TAW2BF
URIm2://K2S5TAW2BF
Kinddefinition
WritingW0 stub
MSC30D30, 30C15
DefinesOrder of a zero or pole
UsesMeromorphic function, Order of a zero, Order of a pole, Laurent series, Holomorphic function
Changed2 November 2026

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