Author |
Topic: Summation of Series using Residue (Read 9351 times) |
|
knightfischer
Junior Member
Gender: 
Posts: 54
|
 |
Summation of Series using Residue
« on: Sep 17th, 2007, 4:28pm » |
 Quote  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:
Posts: 54
|
|
Re: Summation of Series using Residue
« Reply #1 on: Sep 17th, 2007, 4:30pm » |
Quote Modify
|
The assumptions are |f(z)|<=M/|z^k|, k>1, M constants independent of N.
|
|
IP Logged |
|
|
|
Michael Dagg
Senior Riddler
Gender:
Posts: 500
|
|
Re: Summation of Series using Residue
« Reply #2 on: Sep 29th, 2007, 7:25pm » |
Quote Modify
|
Did you solve this problem?
|
|
IP Logged |
Regards,
Michael Dagg
|
|
|
knightfischer
Junior Member
Gender:
Posts: 54
|
|
Re: Summation of Series using Residue
« Reply #3 on: Sep 30th, 2007, 7:56am » |
Quote Modify
|
No. I could not figure it out and no one seemed interested in replying. Can you help?
|
|
IP Logged |
|
|
|
Obob
Senior Riddler
Gender:
Posts: 489
|
|
Re: Summation of Series using Residue
« Reply #4 on: Sep 30th, 2007, 12:14pm » |
Quote 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:
Posts: 1261
|
|
Re: Summation of Series using Residue
« Reply #5 on: Sep 30th, 2007, 1:08pm » |
Quote 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:
Posts: 54
|
|
Re: Summation of Series using Residue
« Reply #6 on: Oct 1st, 2007, 4:15am » |
Quote 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:
Posts: 54
|
|
Re: Summation of Series using Residue
« Reply #7 on: Oct 1st, 2007, 8:40am » |
Quote 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 |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print