Meromorphic functions: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Meg Taylor
m (spelling: wiht -> with)
imported>Meg Taylor
No edit summary
Line 6: Line 6:
Many of [[special function]]s are meromorphic.
Many of [[special function]]s are meromorphic.


For example, [[factorial]] is meromorphic in the whole comples plane (it has [[countable set]] of poles at the negative integer values of the arguemnt), but [[logarithm]] is not, because it has cutline at the negative part of the real axis. However, the eame logarithm becomes  
For example, [[factorial]] is meromorphic in the whole complex plane (it has [[countable set]] of poles at the negative integer values of the argument), but [[logarithm]] is not, because it has cutline at the negative part of the real axis. However, the same logarithm becomes  
meromorphic being considered on the domain of numbers with positive real part.
meromorphic being considered on the domain of numbers with positive real part.

Revision as of 10:05, 10 October 2013

This article is developing and not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek “meros” (μέρος), meaning part, as opposed to “holos” (ὅλος), meaning whole.)

In particular, every holomorphic function can be considered as meromorphic.

Many of special functions are meromorphic.

For example, factorial is meromorphic in the whole complex plane (it has countable set of poles at the negative integer values of the argument), but logarithm is not, because it has cutline at the negative part of the real axis. However, the same logarithm becomes meromorphic being considered on the domain of numbers with positive real part.