# 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