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        general >> complex analysis >> Meromorphic function
        
(Message started by: Tiox on Nov 8^th, 2005, 7:45pm)

Title: Meromorphic function
Post by Tiox on Nov 8^th, 2005, 7:45pm

Please help me a little with this problem:

Show that any function which is meromorphic in the extended complex plane is a rational function.

Title: Re: Meromorphic function
Post by Icarus on Nov 9^th, 2005, 6:41pm

Because meromorphic functions do not have any essential singularities, no sequence of poles can converge to any point within the domain of the function, as such a point would be an essential singularity. But in the extended complex plane (also known as the Riemann Sphere), every infinite set has convergent subsequences (The Reimann Sphere is "compact"). Therefore the set of poles of a function meromorphic on the entire Riemann Sphere must be finite.

Now consider what happens when you remove the poles...

Title: Re: Meromorphic function
Post by Michael_Dagg on Nov 25^th, 2005, 9:50am

Nice explanation Icarus.

If you multiply off the poles to get an entire function which has a limit, possibly infinite, at infinity, and then by using a generalization of Liouville one can infer that this entire function is a polynomial.

Title: Re: Meromorphic function
Post by Icarus on Nov 26^th, 2005, 12:03pm

More specifically, to be meromorphic in the extended plane means that oo is also at most a pole...