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 April 18th, 2011, 05:18 PM #1 Newbie   Joined: Mar 2011 Posts: 26 Thanks: 0 Argument principle problem Let $f$ be meromorphic with no poles or zeroes in $A=\{ z=x+iy|x\in Z\}\cup \{ z=x+iy|y\in Z\}$. If $f(z)=f(z+1)=f(z+i)$ (when these expressions are defined), show that for every $m,n\in Z$ the number of poles and zeroes of $f$ inside the rectangle with vertices at $0,m, in, m+in$ is equal. I am stuck on how to use the fact that $f(z)=f(z+1)=f(z+i)$ in order to show that $\oint_{\Gamma}\frac{f'}{f}=0$ (where $\Gamma$ is the rectangle) so I can use the argument principle to show what I need. I need direction here.
 April 20th, 2011, 10:28 AM #2 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Argument principle problem SonicYouth, Consider the integral of f over a fundamental rectangle. The integral along the opposing sides will cancel by the periodicity, which implies that the sum of the residues of f must be zero. By the way, this implies that f (and in general, any elliptic function) cannot have a solitary simple pole (in a period parallelogram). Now, note that any pole or zero of f is a simple pole of f'/f. Ask yourself whether f'/f is also doubly periodic with the same periods, and then use this to conclude the problem by integrating the logarithmic integral about the period rectangle. -Ormkärr-
April 23rd, 2011, 06:51 AM   #3
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Re: Argument principle problem

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 Originally Posted by Ormkärr SonicYouth, Consider the integral of f over a fundamental rectangle. The integral along the opposing sides will cancel by the periodicity, which implies that the sum of the residues of f must be zero. By the way, this implies that f (and in general, any elliptic function) cannot have a solitary simple pole (in a period parallelogram). Now, note that any pole or zero of f is a simple pole of f'/f. Ask yourself whether f'/f is also doubly periodic with the same periods, and then use this to conclude the problem by integrating the logarithmic integral about the period rectangle. -Ormkärr-
Thanks.

Problem solved

 April 24th, 2011, 01:09 PM #4 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Argument principle problem Glad to be of help. The subject of doubly periodic functions in the complex plane is a vast and rich one, with lots of fascinating developments. I think there is a section devoted to them in Ahlfors, certainly he touches on them. These functions are also called elliptic functions, the name you have almost certainly heard of, and their study goes back a way, with many of the seminal results coming from Abel, but also to Weierstrass and later mathematicians. If you get so far, you should certainly try to study Riemann surfaces, which is a related field, this is a subject that binds together some of the most beautiful parts of all of mathematics! -Ormkärr-
May 7th, 2011, 06:02 PM   #5
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Re: Argument principle problem

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 Originally Posted by Ormkärr If you get so far, you should certainly try to study Riemann surfaces, which is a related field, this is a subject that binds together some of the most beautiful parts of all of mathematics! -Ormkärr-

Funny you mentioned that, but we just started this topic this week. I was actually wondering about something the professor mentioned, he said Riemann surfaces are a "20th century idea" but I thought Riemann died in the 1800's? so was it his idea that was later developed by someone else?

 May 7th, 2011, 09:56 PM #6 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Argument principle problem SonicYouth, Yes, I think that Riemann developed the idea of extending the domain of complex functions to make them bona-fide functions (i.e. remove the ambiguity of multi-valuedness), but I am not sure how far he took these ideas, although he did make important insights into the field which are still seminal results today. I am not wholly familiar with the historical development of mathematics, so I hesitate to state this with complete conviction. I think what your professor means is that the notion of Riemann surface has developed most rapidly in the twentieth century, because by this time, the details of setting complex analysis on rigorous footing had been mostly worked out, so Riemann's own intuitions, for example, had been confirmed by proof. A lot of the developments in theory of Riemann surfaces is related to topology, differential structure, and of course algebraic geometry. These fields themselves were only in nascent phases of development during Riemann's time. It is worth noting that Riemann died at the age of 39, such was his intensity as a mathematician! -Ormkärr-

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