What does the $n$ do in $1/(j \omega)^n$ I'm currently working through a research paper on beam forming. In this paper a magnitude compensation is introduced to compensate for frequency dependency. Due to other calculations low frequencies are lower in magnitude than high frequencies.

To compensate for the frequency dependency they multiply the signal with: 
$$
 \frac{1}{(j \omega)^n}
$$

I know when $n = 1$, the formula is just a simple integrator. But I don't understand what happens when $n$ is higher, and how I should implement it in MATLAB.
What do you mean with Beam in this context.... Particle Beams in LINACs? Structural Beams in Buildings?. 

$j\omega$ is a phasor term from frequency analysis. 

$\frac{1}{j\omega}$ is the integration as @msm showed. 

If you want to integrate two times, just use cumtrapz two times.

But as you said, this is only a "term" complementing a frequency model, which it would be really good to have here in order to help you more... You never simulate that term alone.