Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 35 Location : Right behind you
 Subject: [SOLVED] Complex[2] Tue Jan 05, 2010 10:38 am  
 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 Subject: Re: [SOLVED] Complex[2] Wed 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. Welcome back to the forum Mo! 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 35 Location : Right behind you
 Subject: Re: [SOLVED] Complex[2] Thu Jan 07, 2010 12:14 am  
 Almost done! Sol 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 Subject: Re: [SOLVED] Complex[2] Thu 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! I thought it was a bit too easy for a Mohammad problem! It should have rung a bell. Very cool problem in retrospect though. 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 35 Location : Right behind you
 Subject: Re: [SOLVED] Complex[2] Thu Jan 07, 2010 10:48 am  
 When you have time please move solved problems to the other section 

nick Euclid
Posts : 95 Join date : 20090915 Age : 56 Location : Alexandria
 Subject: Re: [SOLVED] Complex[2] Thu 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... 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 Subject: Re: [SOLVED] Complex[2] Mon 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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 Subject: Re: [SOLVED] Complex[2]  
 
