Editing Zeros and poles (section)

==Function on a curve==

The concept of zeros and poles extends naturally to functions on a ''complex curve'', that is [[complex analytic manifold]] of dimension one (over the complex numbers). The simplest examples of such curves are the [[complex plane]] and the [[Riemann surface]]. This extension is done by transferring structures and properties through [[Atlas (topology)|chart]]s, which are analytic [[isomorphism]]s.

More precisely, let {{mvar|f}} be a function from a complex curve {{mvar|M}} to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point {{mvar|z}} of {{mvar|M}} if there is a chart <math>\phi</math> such that <math> f \circ \phi^{-1}</math> is holomorphic (resp. meromorphic) in a neighbourhood of <math>\phi(z).</math> Then, {{mvar|z}} is a pole or a zero of order {{mvar|n}} if the same is true for <math>\phi(z).</math>

If the curve is [[compact space|compact]], and the function {{mvar|f}} is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros. This is one of the basic facts that are involved in [[Riemann–Roch theorem]].