Tuesday, 23 August 2016

sequences and series - Help with an infinite sum of exponential terms?



I've been trying to calculate the mean squared displacement of a particle confined to a one-dimensional box, and I managed to get an answer in terms of an infinite series of the basic form
n=1(1)nn2exp(an2).
I can't figure out if this series has a simple solution; I can't find it in any table of series, nor can I seem to expand the exponential and collect like terms without running into (1)2 terms. Does this have a solution or is this as far as I can go? Thanks in advance.


Answer



By using the Jacobian Theta function the resulting series can be placed into the form
f(a)=n=1(1)nn2ean2=a2π21212a0θ(12,iuπ)du.



This is developed by the following. Differentiate f(a) to obtain
f(a)=n=1(1)n+1ean2.

Now,
θ(x,it)=1+2n=1eπtn2cos(2nπx)
which yields
f(a)=12(1θ(12,iaπ)).
Integrating with respect to a yields
f(a)=a212a0θ(12,iuπ)du+c0
Since f(0)=12ζ(2) then the presented result is obtained.


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