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.

Friday, May 13, 2011

Imaginary numbers are silly

So, the imaginary constant is a pretty silly constant. While there is nothing wrong with them, they're defined in a very awkward fashion. A much more elegant way of doing it is by using 2x2 matrices.

Define the imaginary constant to be
\[
i=\left(\begin{matrix}0 & -1\\
1 & 0\end{matrix}\right)
\]
which is the rotation matrix for a 90 degree clockwise rotation of a two dimensional vector.

Let's further implicitly put an identity matrix next to all numbers. For example, consider a complex number
\[
z=a+bi=\left(\begin{matrix}a & 0\\
0 & a\end{matrix}\right)+\left(\begin{matrix}0 & -b\\
b & 0\end{matrix}\right)=\left(\begin{matrix}a & -b\\
b & a\end{matrix}\right)
\]
It's trivial to show that this obeys all that we'd expect from complex algebra (and complex conjugate is the same as transposing the matrix). Wasn't that nice? No obscure voodoo mathematics. Just a simple 2x2 matrix. We don't need to justify the existence of matrices. They are not just some ugly trick that's used to simplify calculations.