Editing Zeros and poles (section)

==At infinity==
A function <math> z \mapsto f(z)</math> is ''meromorphic at infinity'' if it is meromorphic in some neighbourhood of infinity (that is outside some [[disk (mathematics)|disk]]), and there is an integer {{mvar|n}} such that 
:<math>\lim_{z\to \infty}\frac{f(z)}{z^n}</math>
exists and is a nonzero complex number.

In this case, the [[point at infinity]] is a pole of order {{mvar|n}} if {{math|''n'' > 0}}, and a zero of order <math>|n|</math> if {{math|''n'' < 0}}.

For example, a [[polynomial]] of degree {{mvar|n}} has a pole of degree {{mvar|n}} at infinity.

The [[complex plane]] extended by a point at infinity is called the [[Riemann sphere]].

If {{mvar|f}} is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros. 

Every [[rational function]] is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.