Editing Zeros and poles (section)

== Examples ==

[[File:Pole-order9-infin.png|right|thumb|300px|A polynomial of degree 9 has a pole of order 9 at ∞, here plotted by [[domain coloring]] of the Riemann sphere.]]
* The function
::<math>f(z) = \frac{3}{z}</math>
: is meromorphic on the whole Riemann sphere. It has a pole of order 1 or simple pole at <math> z= 0,</math> and a simple zero at infinity.

* The function
:: <math>f(z) = \frac{z+2}{(z-5)^2(z+7)^3}</math>
: is meromorphic on the whole Riemann sphere. It has a pole of order 2 at <math> z=5,</math> and a pole of order 3 at <math> z = -7</math>. It has a simple zero at <math> z=-2,</math> and a quadruple zero at infinity.

* The function
:: <math>f(z) = \frac{z-4}{e^z-1}</math>
: is meromorphic in the whole complex plane, but not at infinity. It has poles of order 1 at <math> z=2\pi ni\text{ for } n\in\mathbb Z</math>. This can be seen by writing the [[Taylor series]] of <math> e^z</math> around the origin.

* The function
::<math>f(z) = z</math>
: has a single pole at infinity of order 1, and a single zero at the origin.

All above examples except for the third are [[rational functions]]. For a general discussion of zeros and poles of such functions, see {{sectionlink|Pole–zero plot|Continuous-time systems}}.