1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Parametrized integral and sum

  1. Jun 4, 2015 #1
    1. The problem statement, all variables and given/known data
    The story is that I would like to evaluate the coinciding point limit (when ## (x^0, x^1)→(y^0,y^1)##) of these two terms:

    \begin{eqnarray*}
    &&\frac{1}{2L}e^{\frac{i}{2}eE\left((x ^1)^2-(y^1)^2\right)}\left( im\left( x^0-y^0+ x^1-y^1\right) \right) \sum_{n=0}^{\infty}e^{-ik_n( x^0-y^0+x^1-y^1)}\\
    &&\times \frac{1}{L}\int_{-L/2}^{L/2}dx\left( -1\right)^{n}\cos \left( eE\left(\frac{L^2}{4} -x^2\right) +\left( 2n+1\right) \pi \frac{x}{L}\right) \\
    &&+\frac{1}{2L}e^{\frac{i}{2}eE\left((x ^1)^2-(y^1)^2\right)}m \sum_{n=0}^{\infty} e^{-ik_n( x^0-y^0+x^1-y^1)}\\
    &&\times \frac{1}{L}\int_{-L/2}^{L/2}dx\left(-1 \right)^{n}\left(2x-L \right)\sin\left( eE\left(\frac{L^2}{4}-x^2 \right)+\left(2n+1\right)\pi \frac{x}{L} \right)
    \end{eqnarray*}

    m, e, E, L are respectively the mass, charge, electric field strength, length of the box, and are constant. The energy levels ##k_n##'s are given by ##k_n=(n+1/2)\frac{\pi}{L}##.




    2. Relevant equations & 3. The attempt at a solution

    1. There will be no singularities since another calculation of mine has shown that.

    2. When I put ## \frac{1}{L}\int_{-L/2}^{L/2}dx\left( -1\right)^{n}\cos \left( eE\left(\frac{L^2}{4} -x^2\right) +\left( 2n+1\right) \pi \frac{x}{L}\right) ## in Mathematica, I got something that decays like ##\frac{1}{n}##, when ##n## is large.

    So I expect the sum will give me ##\log (1-exp...)## (appropriate dissipative term ##\pm i \epsilon## may be needed); of course there will be some residue.

    3. My burning question is: is there an analytic way to expand the integral which depends on ##n## as ##\frac{C}{n}+D+...##, for some constant ##C, D##, which are to be found?

    Thanks!
     
  2. jcsd
  3. Jun 4, 2015 #2

    Zondrina

    User Avatar
    Homework Helper

    The limit for the first term should be obvious from inspection.

    The limit for the second term has a divergent series ##\sum_{n=0}^{\infty} 1##.

    The integrals look like monsters to do analytically (although it appears possible), why not just observe the behaviour of the other terms and use some simple limit laws?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Parametrized integral and sum
  1. Sum to Integral (Replies: 2)

  2. Integral to sum (Replies: 2)

  3. Parametric Integral? (Replies: 1)

Loading...