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.

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