Math 2.0

Functions of a complex variable Definition

Zeros and poles

Builds on Meromorphic functionZero of a holomorphic functionPoleHolomorphic function

Definition

Let U⊆ℂU \subseteq \C be an open set and let ff be a meromorphic function on UU. So there is a closed discrete set P⊂UP \subset U such that ff is holomorphic on U∖PU \setminus P and each point of PP is a pole of ff.

  1. A zero of ff is a point a∈U∖Pa \in U \setminus P with f(a)=0f(a) = 0, as in the definition of a zero of a holomorphic function.
  2. A pole of ff is a point a∈Pa \in P, that is, an isolated singularity with ∣f(z)∣→∞\abs{f(z)} \to \infty as z→az \to a, as in the definition of a pole.

Remarks

Example: f(z)=z−1z2+1f(z) = \dfrac{z - 1}{z^2 + 1} on ℂ\C has one zero, at 11, and two poles, at ii and −i-i, where the denominator vanishes and the numerator does not. Non-example: e1/ze^{1/z} is not meromorphic on ℂ\C, and 00 is neither a zero nor a pole of it. The point 00 is an essential singularity: near 00, e1/ze^{1/z} is neither bounded nor tends to ∞\infty. If UU is connected and ff is not identically 00, the zeros of ff are isolated and the poles of ff are the zeros of 1/f1/f. The definition extends to meromorphic functions on the Riemann sphere or on a Riemann surface, using local coordinates. A nonzero rational function, seen on the Riemann sphere, has as many zeros as poles when both are counted with their orders.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
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 singularitydefinitionZero of a holomorphic function, definition, no evidence levelZero of a holomorphic…definitionPole, definition, no evidence levelPoledefinitionMeromorphic function, definition, no evidence levelMeromorphic functiondefinitionZeros and poles, definition, no evidence levelZeros and polesdefinition
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.
  • no evidence level
  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. Zero of a holomorphic function definition
  14. Pole definition
  15. Meromorphic function definition
  16. Zeros and poles definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Zeros_and_poles
About this page
Tag9DNDF7AX59
URIm2://9DNDF7AX59
Kinddefinition
WritingW0 stub
MSC30D30, 30C15
DefinesZeros and poles
UsesMeromorphic function, Zero of a holomorphic function, Pole, Holomorphic function
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.