Riemann Sum: K(t) = c^(3/2)/(m^(1/2)(N-1)) Sum(k=1, N-1)....

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komizuki

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\(\displaystyle \L\\\lim_{N\to\infty}\) for the function:

\(\displaystyle \L\\K(t) = \frac{c^{\frac{3}{2}}}{m^{\frac{1}{2}}(N-1)}\sum_{k=1}^{N-1} q_{k} Sin^{2}[q_{k}M]e^{-(\frac{c}{m})^{\frac{1}{2}}q_{k}t}\)

where \(\displaystyle \L\\q_{k} =\frac{{\pi}k}{N} + O(\frac{1}{N^{2}}\)), \(\displaystyle \L\\M = aN\) with \(\displaystyle \L\\0<a<1\) and c,m are just constants (can be set to 1 if you like)

how does K(t) look like in Integralform (limit riemann sum -> riemann integral)???

The following tries involve setting the constants (m,c) to 1:
I have already tried running this with mathematica (no result).
I set a to some arbitrary values (.5, .75, .95) no results.
if I leave out the t in the equation I get values, but that doesn't help me, since I am supposed to get something ~ 1/t^2 !!!
I do not know any further way, with the definition of the riemann integral how to solve that.
please help!
 
order of, that just means, that in the expansion of q_k the later terms are less meaningfull in the limit (terms with higher power of N in 1/N^x contribute less).
sry but if you do not know that, then i don't think you can help me...
 
Why isn't anyone capable of answering that. my prof said this is school-stuff. so easy a kid could do it (of course he always says that no matter how many pages/days i need). i mean i know the definition of the riemann sum (here:)

lim_{N->\infty} \frac{1}{N} \sum_{k=1}^{N} f(q_k,t) = \frac{1}{\pi}\int_{0}^{\pi}f(q,t)dq (done here for a \pi periodic function)

example my prof gave me. problem i have, my function is not \pi periodic at all, because of the exponential the sine squared would fullfill that requirement.
 
\(\displaystyle \L O\(f\(n\)\)\) is a set of functions. How in the world is someone supposed to sum up a bunch of sets? Is this for an elementary Calculus class?
 
nobody is supposed to sum that up (the O()) i just mentioned it. it is supposed to be left out because 1/N^2 goes faster to zero then the first term. so the problem IS NOT the O(anything) i just mentioned it and probably should have just left it out in order to make it clearer.
no its not for an elementary calculus class but a small problem in my physics phd.
 
Disregarding the constants c and m and the big-O, the problem is still not trivial, but I have seen similar problems done.

Comming from an analysis approach, you must discover a closed form for your partial sum \(\displaystyle \L S_n\) for your sequence. Then take the limit of \(\displaystyle S_n\). Its tedius to break down something like \(\displaystyle S_4\) and then relate it to \(\displaystyle S_1, S_2, S_3\) but it can be done.

With the simplified sum below:

\(\displaystyle \L S_n\(t\) \,\, = \,\, \frac{\pi}{N^2-N} \sum_{k=1}^{N-1}k\frac{\sin^2\(ka \pi\)}{e^{\frac{\pi k}{N}t}\)

It must be then that the limit of only the sum is \(\displaystyle \frac{\(N^2-N\)}{t^2\pi}\) in order to get your desired limit of \(\displaystyle \frac{1}{t^2}\).
 
thank you for the approach i will try it on my train travel... (and get results hopefully by tomorrow). with trivial yes exactly i don't think it is, but my prof. always says its trivial *g
 
komizuki said:
Why isn't anyone capable of answering that.
Click to expand...
After looking at it once, I concluded it most likely had something to do with physics. You have now confirmed that. I don’t do physics or any other form of applied mathematics. Could you not find a physics forum? Surely such exists.
 
the problem with a physics forum is:

i don't need an explanation why anything moves in a certain way because of a certain reason. i don't think many people would even barely understand my problem (it has to do with a lagrangian with bilinear coupling to a bath of harmonic oscillators which describes a tunneling in a linear chain via path integral formalism). the problem i have is hence not a physical, but a pure mathematical:

riemann sum lim N-> infinity TO riemann integral
 
as i said it is NOT A PHYSICS problem but a MATH PROBLEM.
definition of the riemann integral via lim of the riemann sum every mathematician should be able to do that and hence help me with my problem.
 
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