wu :: forums
wu :: forums - Summation of Series using Residue

Welcome, Guest. Please Login or Register.
Mar 28th, 2023, 4:06am

RIDDLES SITE WRITE MATH! Home Home Help Help Search Search Members Members Login Login Register Register
   wu :: forums
   general
   complex analysis
(Moderators: Icarus, Grimbal, william wu, ThudnBlunder, towr, Eigenray, SMQ)
   Summation of Series using Residue
« Previous topic | Next topic »
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print
   Author  Topic: Summation of Series using Residue  (Read 9116 times)
knightfischer
Junior Member
**





   


Gender: male
Posts: 54
Summation of Series using Residue  
« on: Sep 17th, 2007, 4:28pm »
Quote Quote Modify Modify

In the proof that  
Sum (n=-inf to +inf) of {f(n)} = -{sum residues pi(cot(pi(z)))f(z) at poles of f(z)},
the square  Cn, with verices at (N+1/2)(1+i) is used as a path.  The proof for f(z) with a finite number of poles is straightforward using the residue theorem.  However, it is not clear to me how to extend this to f(z) with infinite number of poles.  The book says "we can obtain the required result by an appropriate limiting procedure."  With f(z) having a finite number of poles, Cn is chosen with N large enough to enclose all the poles.  If f(z) has an infinite number of poles, there is no N large enough.  How can the residue theorem be applied?
 
Any help would be appreciated.
IP Logged
knightfischer
Junior Member
**





   


Gender: male
Posts: 54
Re: Summation of Series using Residue  
« Reply #1 on: Sep 17th, 2007, 4:30pm »
Quote Quote Modify Modify

The assumptions are |f(z)|<=M/|z^k|, k>1, M constants independent of N.
IP Logged
Michael Dagg
Senior Riddler
****






   


Gender: male
Posts: 500
Re: Summation of Series using Residue  
« Reply #2 on: Sep 29th, 2007, 7:25pm »
Quote Quote Modify Modify

Did you solve this problem?
IP Logged

Regards,
Michael Dagg
knightfischer
Junior Member
**





   


Gender: male
Posts: 54
Re: Summation of Series using Residue  
« Reply #3 on: Sep 30th, 2007, 7:56am »
Quote Quote Modify Modify

No.  I could not figure it out and no one seemed interested in replying.  Can you help?
IP Logged
Obob
Senior Riddler
****





   


Gender: male
Posts: 489
Re: Summation of Series using Residue  
« Reply #4 on: Sep 30th, 2007, 12:14pm »
Quote Quote Modify Modify

I'm not really clear on what your assumption is in the second post.  The assumption |f(z)|<=M/|z^k| implies that f has only a single pole.
IP Logged
Sameer
Uberpuzzler
*****



Pie = pi * e

   


Gender: male
Posts: 1261
Re: Summation of Series using Residue  
« Reply #5 on: Sep 30th, 2007, 1:08pm »
Quote Quote Modify Modify

I don't understand the wordings either, but for a square area and infinite poles don't you take the integrals on the sides and then let the limit of N go to infinity? That should take care of your infinite poles.  
 
IP Logged

"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple

Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
knightfischer
Junior Member
**





   


Gender: male
Posts: 54
Re: Summation of Series using Residue  
« Reply #6 on: Oct 1st, 2007, 4:15am »
Quote Quote Modify Modify

Sorry I was not clear.  The assumptions are: along the path Cn, |f(z)|<=M/|z^k|, where k>1 and M are constants independent of N.
IP Logged
knightfischer
Junior Member
**





   


Gender: male
Posts: 54
Re: Summation of Series using Residue  
« Reply #7 on: Oct 1st, 2007, 8:40am »
Quote Quote Modify Modify

If there are an infinite number of poles, then for any N, there could still be poles outside of the square Cn.  If N goes to infinity, can you still assume at some point Cn encloses all the poles, where there are an infinite number of poles, any one of which could be an infinite distance from the origin?  It is not clear that the Residue Theorem can be applied unless you can prove that Cn encloses all the poles.
IP Logged
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print

« Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright 2000-2004 Yet another Bulletin Board