Strange formulation of Gauss' Theorem

[Ssnow]

TL;DR Summary

Is this a complicated expression for the ordinary Gauss theorem for the flux of the electric field ?

Hi to all!
The ordinary Gauss theorem states that $\Phi\left(\vec{E}\right)\,=\, \frac{\sum_{i=1}^{n}q_{i}}{\varepsilon_{0}}$ where $\sum_{i=1}^{n}q_{i}$ is the sum of all charges internal of a closed surface and $\varepsilon_{0}$ is the dielectric constant in the empty. Now I ask to the PF if this formula:

$\Phi\left(\vec{E}\right)\,=\, sign{\left(\sum_{i=1}^{n}q_{i}\right)}\cdot \left(\int_{-\infty}^{+\infty}e^{-\frac{\pi\varepsilon_{0}}{\left|\sum_{i=1}^{n}q_{i}\right|}y^2}dy\right)^2$

is equivalent to the previous and if it is mathematically correct.
Thank you!
Ssnow

 

[thelaxiankey]

It looks good to me; the integral is called a Gaussian integral and if you evaluate it, you get the right answer. I'm wondering where you saw it formulated like this?

 

[vanhees71]

And for what should it be good for?

 

[Ssnow]

In fact it seems not very useful to write le Gauss theorem in this form...
the funny thing is that this formulation relates two result associated to the name of Gauss, respectively the Gauss theorem and the Gaussian integral ...
I encountered this formulation because I am writing a compendium on the Gaussian integral and its generalizations, so I think to put also this nice example ...
Ssnow