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 [SOLVED] Complex[2]

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Mohammad
Descartes
Descartes
Mohammad

Posts : 100
Join date : 2009-11-05
Age : 35
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[SOLVED] Complex[2] Empty
PostSubject: [SOLVED] Complex[2]   [SOLVED] Complex[2] EmptyTue Jan 05, 2010 10:38 am

Back again Exclamation
problem bounce
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Bruno
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Bruno

Posts : 184
Join date : 2009-09-15
Age : 32
Location : the infinite, frictionless plane of uniform density

[SOLVED] Complex[2] Empty
PostSubject: Re: [SOLVED] Complex[2]   [SOLVED] Complex[2] EmptyWed Jan 06, 2010 5:46 pm

The function is completely determined by its values on the parallelogram with vertices w_1 and w_2, and only takes the values it takes there. Since the function is entire, it has no singularities on the parallelogram; hence it is bounded there, and hence everywhere, and by Liouville's theorem a bounded entire function is constant.

What a Face

Welcome back to the forum Mo!
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Mohammad
Descartes
Descartes
Mohammad

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

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PostSubject: Re: [SOLVED] Complex[2]   [SOLVED] Complex[2] EmptyThu Jan 07, 2010 12:14 am

Almost done!
Sol bounce
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Bruno
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Bruno

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PostSubject: Re: [SOLVED] Complex[2]   [SOLVED] Complex[2] EmptyThu Jan 07, 2010 10:13 am

Oh yeah! For some reason I had read the problem as w_1/w_2 being in C - R. So I only took care of the first part of the problem...

I should pay more attention! Mad I thought it was a bit too easy for a Mohammad problem! It should have rung a bell. Basketball

Very cool problem in retrospect though.
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Mohammad
Descartes
Descartes
Mohammad

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

[SOLVED] Complex[2] Empty
PostSubject: Re: [SOLVED] Complex[2]   [SOLVED] Complex[2] EmptyThu Jan 07, 2010 10:48 am

When you have time please move solved problems to the other section bounce
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nick
Euclid
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PostSubject: Re: [SOLVED] Complex[2]   [SOLVED] Complex[2] EmptyThu Feb 04, 2010 12:51 pm

So,

Roughly speaking, a periodical analytic entire function can not have more than one period ?

Complex analysis looks to me like sculpture...
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Bruno
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PostSubject: Re: [SOLVED] Complex[2]   [SOLVED] Complex[2] EmptyMon Feb 08, 2010 9:26 pm

That is correct. A doubly periodic entire function is necessarily constant. However there are some very interesting doubly periodic meromorphic functions (the elliptic functions).
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PostSubject: Re: [SOLVED] Complex[2]   [SOLVED] Complex[2] Empty

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