# I Integral with complex oscillating phase

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1. Nov 16, 2016

### anthony2005

Does there exist and analytical expression for the following integral?

$$I\left(s,m_{1},m_{2},L\right)=\sum_{\boldsymbol{n}\in\mathbb{N}^{3}\backslash\left\{ \boldsymbol{0}\right\} }\int\frac{d^{3}q}{\left(2\pi\right)^{3}}\frac{1}{2\omega_{1}\left(\boldsymbol{q}\right)\omega_{2}\left(\boldsymbol{q}\right)}\frac{\omega_{1}\left(\boldsymbol{q}\right)+\omega_{2}\left(\boldsymbol{q}\right)}{s-\left(\omega_{1}\left(\boldsymbol{q}\right)+\omega_{2}\left(\boldsymbol{q}\right)\right)^{2}}e^{iL\boldsymbol{q}\cdot\boldsymbol{n}}$$

where $s,m_{1},m_{2},L>0$ and $\omega_{1,2}\left(\boldsymbol{q}\right)=\sqrt{|\boldsymbol{q}|^{2}+m_{1,2}^{2}}$.

Also a series expansion is ok. Indeed, for an easier integral:
$$J\left(m,L\right)=\sum_{\boldsymbol{n}\in\mathbb{N}^{3}\backslash\left\{ \boldsymbol{0}\right\} }\int\frac{d^{3}q}{\left(2\pi\right)^{3}}\frac{1}{2\omega\left(\boldsymbol{q}\right)}e^{iL\boldsymbol{q}\cdot\boldsymbol{n}}=\frac{m}{4\pi^{2}L}\sum_{\boldsymbol{n}\in\mathbb{N}^{3}\backslash\left\{ \boldsymbol{0}\right\} }\frac{K_{1}\left(|\boldsymbol{n}|mL\right)}{|\boldsymbol{n}|}$$

where $K$ is the modified Bessel function.

The integral becomes one-dimensional in spherical coordinates, but still I have not found an analytical result in the literature.

2. Nov 21, 2016

### Greg Bernhardt

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.