Tuesday, May 31, 2011

Neat function

Stumbled on this neat function that acts like a singularity, except drops to zero instead of diverging within a neighborhood dependent on a parameter $\delta$.

$x^{-n}e^{-\frac{1}{2}\left(\frac{\delta}{x}\right)^{2n}}\approx\begin{cases} x^{-n} & |x|>\delta\\ 0 & |x|<\delta \end{cases}$

For $\delta > 0$, $n\in\mathbb{N}_{1}$. I imagine this may help with unstable simulations of inverse square law type interaction.

Integral is really nice too (for n > 2), by diverging for small δ, means it could possibly used for renormalization work?
$\int_0^{\infty} dx\ x^{-n}e^{-\frac{1}{2}\left(\frac{\delta}{x}\right)^{2n}}\sim\delta^{-n+1}$

See WA for some plots.

1 comment:

1. I'm an Open Source Programmer with a MSc computer science. I was Stumbled on this neat function that acts like a singularity.