Meromorphic function: Difference between revisions

Revision as of 00:17, 22 November 2023

Revision as of 16:02, 27 November 2023

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==Properties==

Since the poles of a meromorphic function are isolated, there are at most [[countable|countably]] many for a meromorphic function.<ref name=Lang_1999/> The set of poles can be infinite, as exemplified by the function <math display="block">f(z) = \csc z = \frac{1}{\sin z}.</math>

By using [[analytic continuation]] to eliminate [[removable singularity|removable singularities]], meromorphic functions can be added, subtracted, multiplied, and the quotient <math>f/g</math> can be formed unless <math>g(z) = 0</math> on a [[connected space|connected component]] of ''D''. Thus, if ''D'' is connected, the meromorphic functions form a [[field (mathematics)|field]], in fact a [[field extension]] of the [[complex numbers]].