# Parametrized integral and sum

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1. Jun 4, 2015

### Willi Tschau

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. Jun 4, 2015

### Zondrina

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?