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wu :: forums    general    complex analysis (Moderators: Eigenray, william wu, towr, ThudnBlunder, Grimbal, SMQ, Icarus)    Meromorphic function « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Meromorphic function  (Read 7568 times) Tiox Guest Meromorphic function   « on: Nov 8th, 2005, 7:45pm » Quote Modify Remove Please help me a little with this problem:   Show that any function which is meromorphic in the extended complex plane is a rational function. IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 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 "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Michael Dagg Senior Riddler     Gender: Posts: 500 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. IP Logged Regards, Michael Dagg Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 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... IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth => complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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