Editing Zeros and poles (section)

== Definitions ==
A [[function of a complex variable]] {{mvar|z}} is [[Holomorphic function|holomorphic]] in an [[open set|open domain]] {{mvar|U}} if it is [[differentiable function|differentiable]] with respect to {{mvar|z}} at every point of {{mvar|U}}. Equivalently, it is holomorphic if it is [[analytic function|analytic]], that is, if its [[Taylor series]] exists at every point of {{mvar|U}}, and converges to the function in some [[neighbourhood (mathematics)|neighbourhood]] of the point. A function is [[meromorphic function|meromorphic]] in {{mvar|U}} if every point of {{mvar|U}} has a neighbourhood such that at least one of {{mvar|f}} and {{math|1/''f''}} is holomorphic in it.

A '''[[zero of a function|zero]]''' of a meromorphic function {{mvar|f}} is a complex number {{mvar|z}} such that {{math|1=''f''(''z'') = 0}}. A '''pole''' of {{mvar|f}} is a zero of {{math|1/''f''}}.

If {{mvar|f}} is a function that is meromorphic in a neighbourhood of a point <math>z_0</math> of the [[complex plane]], then there exists an integer {{mvar|n}} such that 
:<math>(z-z_0)^n f(z)</math>
is holomorphic and nonzero in a neighbourhood of <math>z_0</math> (this is a consequence of the analytic property).
If {{math|''n'' > 0}}, then <math>z_0</math> is a ''pole'' of '''order''' (or multiplicity) {{mvar|n}} of {{mvar|f}}. If {{math|''n'' < 0}}, then <math>z_0</math> is a zero of order <math>|n|</math> of {{mvar|f}}. ''Simple zero'' and ''simple pole'' are terms used for zeroes and poles of order <math>|n|=1.</math> ''Degree'' is sometimes used synonymously to order.

This characterization of zeros and poles implies that zeros and poles are [[isolated point|isolated]], that is, every zero or pole has a neighbourhood that does not contain any other zero and pole.

Because of the ''order'' of zeros and poles being defined as a non-negative number {{mvar|n}} and the symmetry between them, it is often useful to consider a pole of order {{mvar|n}} as a zero of order {{math|−''n''}} and a zero of order {{mvar|n}} as a pole of order {{math|−''n''}}. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.

A meromorphic function may have infinitely many zeros and poles. This is the case for the [[gamma function]] (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The [[Riemann zeta function]] is also meromorphic in the whole complex plane, with a single pole of order 1 at {{math|1=''z'' = 1}}. Its zeros in the left halfplane are all the negative even integers, and the [[Riemann hypothesis]] is the conjecture that all other zeros are along {{math|1=Re(''z'') = 1/2}}.

In a neighbourhood of a point <math>z_0,</math> a nonzero meromorphic function {{mvar|f}} is the sum of a [[Laurent series]] with at most finite ''principal part'' (the terms with negative index values):
:<math>f(z) = \sum_{k\geq -n} a_k (z - z_0)^k,</math>
where {{mvar|n}} is an integer, and <math>a_{-n}\neq 0.</math> Again, if {{math|''n'' > 0}} (the sum starts with <math>a_{-|n|} (z - z_0)^{-|n|}</math>, the principal part has {{mvar|n}} terms), one has a pole of order {{mvar|n}}, and if {{math|''n'' ≤ 0}} (the sum starts with <math>a_{|n|} (z - z_0)^{|n|}</math>, there is no principal part), one has a zero of order <math>|n|</math>.