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Tiox
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 Meromorphic function   « on: Nov 8th, 2005, 7:45pm » Quote Modify Remove

Show that any function which is meromorphic in the extended complex plane is a rational function.
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Icarus
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 Re: Meromorphic function   « Reply #1 on: Nov 9th, 2005, 6:41pm » Quote Modify

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...
 « Last Edit: Nov 9th, 2005, 7:12pm by Icarus » IP Logged

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Michael Dagg
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 Re: Meromorphic function   « Reply #2 on: Nov 25th, 2005, 9:50am » Quote Modify

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.
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Michael Dagg
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 Re: Meromorphic function   « Reply #3 on: Nov 26th, 2005, 12:03pm » Quote Modify

More specifically, to be meromorphic in the extended plane means that oo is also at most a pole...
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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