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   A summation of series probalem
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   Author  Topic: A summation of series probalem  (Read 24307 times)
comehome1981
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A summation of series probalem  
« on: Feb 4th, 2009, 10:14am »
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Someone know the value of the following series:
 
sum^{infty}_{k=-infty} k^2 * exp(-k^2/2) , k is an integer
 
I did it as follows:
 
(1)
sum^{N}_{k=-N} k^2 * exp(-k^2/2) = sum^{N}_{k=-N}{ residues of pi*cot(pi*z) f(z) at integers -N, -N+1, .... , 0, 1, ... , N }
 
(2)
int_{C_N} (pi*cot(pi*z ) f(z) ) dz = 2pi*i sum^{N}_{k=-N}{ residues of pi*cot(pi*z ) f(z) at integers -N, -N+1, .... , 0, 1, ... , N } ( By the Residue Theorem )
 
where C_N be a square with vertices at (N+1/2) x (+- 1 +- i )
 
(3)
lim_{N->infty} int_C_N (pi*
cot(pi*z ) f(z) ) dz = 2pi*i (int_{-infty}^{infty} x^2 exp(-x^2/2) dx)  
 
I  have question about the equality in (3)
Why the two integral are equal??
 
Thanks so much for responses.
« Last Edit: Feb 4th, 2009, 10:16am by comehome1981 » IP Logged
Eigenray
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Re: A summation of series probalem  
« Reply #1 on: Feb 4th, 2009, 3:59pm »
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on Feb 4th, 2009, 10:14am, comehome1981 wrote:

Why the two integral are equal??

They're not.  If they were we would have
n2 e-n^2/2 = x2e-x^2/2dx
but the LHS is 2.506627759, while the RHS is {2} = 2.506628275.
 
One problem with your approach is that the integrals of z2 cot(z) e-z^2/2, over the sides of the square, do not converge.  You would want to keep Im z relatively small to avoid e-z^2/2 blowing up.
 
Your sum can be written in terms of the theta function
(t) = e-n^2 t
as -1/ '(1/(2)).  I don't know if there is a closed form for this.  Where did you get this problem?
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comehome1981
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Re: A summation of series probalem  
« Reply #2 on: Feb 5th, 2009, 11:22am »
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Eigenray, thanks for your reply.
 
well, From my work, I just come up with this series from normal distribution somehow.
 
they are really closed in your numerical values, is it possible just because of the error coming from computer? and actually they are equal?
 
 
Or if the following are equal
 
 sum^{infty}_{k=-infty} k^2 * exp(-k^2/2)= sum^{infty}_{k=-infty}  exp(-k^2/2)
 
I need if this two are equal or not?  
 
Anyone knows??
 
« Last Edit: Feb 5th, 2009, 11:27am by comehome1981 » IP Logged
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Re: A summation of series probalem  
« Reply #3 on: Feb 5th, 2009, 12:21pm »
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Unless there is a bug in Mathematica, Maple and PARI/GP they are not equal.
 
It is easy to approximate the sum: let f(x) = x2 e-x^2/2.  Then f is decreasing for x > 2.  Therefore by considering Riemann sums,
EN = k>N f(k) < N  f(x)dx.
 < N x2 e-x N/2 dx
 = 2e-N^2/2 (8+4N2+N4)/N3,
which is < .4*10-9 for N=7.
This means that the sum is approximated to within 10-9 by
k=-77 f(k) ~ 2.506627759.
 
On the other hand, the integral is well known to be {2} ~ 2.506628275.
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comehome1981
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Re: A summation of series probalem  
« Reply #4 on: Feb 6th, 2009, 7:33am »
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Thanks. That helps a lot.
 
I think they are not equal in (3). But is the following ture?
sum^{infty}_{k=-infty} k^2 * exp(-k^2/2)= sum^{infty}_{k=-infty}  exp(-k^2/2)  
 
I have been trying using program to calculate, and they are not equal. Just wonder if there is a closed form for above equation.
« Last Edit: Feb 6th, 2009, 7:34am by comehome1981 » IP Logged
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Re: A summation of series probalem  
« Reply #5 on: Feb 6th, 2009, 1:05pm »
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No they are not equal.  But there is a reason that the sums are so close to the corresponding integrals.
 
e-k^2/2 = 1 + 2e-1/2 + 2e-4/2 + 2e-9/2 + ...
 ~ 1 + 1.2 + 0.27 + 0.022 + 0.00067 + 0.0000075 + ...
converges rapidly enough, but we can make it converge even faster.  Using Poisson summation, one can show that the theta function satisfies
(1/t) = t  (t).
So
e-k^2/2 = (1/(2)) = {2} (2)
But
(2) = 1 + 2exp(-2^2) + 2exp(-8^2) + 2exp(-18^2) + ...
 ~ 1 + 5*10-9 + 10-34 + 10-77 + ...
 = 1.0000000053505759821...
is very close to 1.  So e-k^2/2 differs from e-x^2/2dx by around 10-8.
 
Similarly, let f(x) = x2 e-x^2/2.  Then the Fourier transform of f is
F(t) = f(x) e-2i x t dx
 = {2} (1-42t2) exp(-22t2),
so by Poisson summation,
f(n) = F(n)
 = {2} [ 1 + 2(1-42)exp(-22) + 2(1-162)exp(-82) + 2(1-362)exp(-182) + ... ]
 ~ {2} [ 1 - 2*10-7 - 2*10-32 - 5*10-75 - ... ]
 = {2} * 0.99999979411830293505...
« Last Edit: Feb 6th, 2009, 1:13pm by Eigenray » IP Logged
comehome1981
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Re: A summation of series probalem  
« Reply #6 on: Feb 6th, 2009, 1:13pm »
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Ok, now it is sure that they are not equal.  
 
A bad news for me.  
 
Anyway, really thanks for those explanation.
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comehome1981
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Re: A summation of series probalem  
« Reply #7 on: Feb 6th, 2009, 2:04pm »
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Sorry for another question:
 
Wonder if there exists a function f(x) s.t.
 
sum^{infty}_{k=-infty} x^2 * exp(-x^2/2)*f(x)= sum^{infty}_{k=-infty}  exp(-x^2/2)*f(x)  
 
is it possible to know what is the form of f(x)?
 
 
 
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Re: A summation of series probalem  
« Reply #8 on: Feb 6th, 2009, 2:42pm »
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f(x)=0 is the most obvious candidate.
And further any function where f(0)=0 and f(-x)=-f(x) (so for example, f(x)=x or f(x)=sin(x) )
Aside from functions where both series are 0, I don't think there are any solutions. (But the ones above are probably not the only ones that accomplish this)
« Last Edit: Feb 6th, 2009, 2:42pm by towr » IP Logged

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Re: A summation of series probalem  
« Reply #9 on: Feb 6th, 2009, 4:22pm »
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You can pick f0 to be any function you want (so long as both sides converge) and then there is a unique constant c such that f(x) = f0(x) + c works.
 
Or you can pick f(n) to be whatever you want for n  0 (again, as long the series converge), and then just solve for f(0).  (Or with 0 replaced by any integer, other than 1.)
 
And of course any linear combination of solutions is also a solution.  The space of all functions is infinite dimensional so one equation won't narrow it down much.  But whether there is any "natural" such function I don't know.
« Last Edit: Feb 6th, 2009, 4:24pm by Eigenray » IP Logged
comehome1981
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Re: A summation of series probalem  
« Reply #10 on: Feb 7th, 2009, 12:09pm »
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on Feb 6th, 2009, 2:42pm, towr wrote:
any function where f(0)=0 and f(-x)=-f(x) (so for example, f(x)=x or f(x)=sin(x) )

 
why f(x)=x will do?
 
sum^{infty}_{x=-infty} x^2 * exp(-x^2/2) *x= sum^{infty}_{x=-infty}  exp(-x^2/2) *x   ??  
 
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comehome1981
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Re: A summation of series probalem  
« Reply #11 on: Feb 7th, 2009, 12:33pm »
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on Feb 6th, 2009, 4:22pm, Eigenray wrote:

Or you can pick f(n) to be whatever you want for n  0 (again, as long the series converge), and then just solve for f(0).  (Or with 0 replaced by any integer, other than 1.)
 

I think this is easier.  Thanks
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Re: A summation of series probalem  
« Reply #12 on: Feb 7th, 2009, 1:05pm »
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on Feb 7th, 2009, 12:09pm, comehome1981 wrote:
why f(x)=x will do?
 
sum^{infty}_{x=-infty} x^2 * exp(-x^2/2) *x= sum^{infty}_{x=-infty}  exp(-x^2/2) *x   ??  
Because x3 = - (-x)3, so for the first series the positive and negative k cancel eachother out. And for the second series x = -(-x), so there also the negative and positive k cancel eachother out. So both series are zero, and zero equals zero.
It goes for any pair of odd and even functions.  
 
 
sum^{infty}_{x=-infty} x^2 * exp(-x^2/2) *x
=
sum^{infty}_{x=-infty} x^3 * exp(-x^2/2)  
=
sum^{-1}_{x=-infty} x^3 * exp(-x^2/2)  + 0^2*exp(0) + sum^{infty}_{x=1} x^2 * exp(-x^2/2)
=
sum^{infty}_{x=1} (-x)^3 * exp(-(-x)^2/2)  + sum^{infty}_{x=1} x^2 * exp(-x^2/2)
=
sum^{infty}_{x=1} -x^3 * exp(-x^2/2)  + sum^{infty}_{x=1} x^2 * exp(-x^2/2)
=
-sum^{infty}_{x=1} x^3 * exp(-x^2/2)  + sum^{infty}_{x=1} x^2 * exp(-x^2/2)
=
0
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comehome1981
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Re: A summation of series probalem  
« Reply #13 on: Feb 7th, 2009, 1:19pm »
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yes I got it , thanks
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