# agm-ized symmetric derivative

The symmetric derivative looks like there’s an average in it, so what about changing that average
with something else, like the arithmetic geometric mean:

$\lim_{h\rightarrow 0} \frac{\mathrm{agm}(f(x+h),f(x-h))}{h}$

# Cayley tables of all finite groups of order 576

I’m going to try to put up an uncompressed version somewhere -the file weighs in at 900mb, but it’s interesting.

# observation from a question on m.se

Set $g(z)=\sum_{n=1}^{\infty} \left(\frac{z}{n}\right)^{n}$, then the sum:

$\sum_{m=1}^{\infty} \frac{m!}{m^{m}}$ can be interpreted as the value of the Laplace

Transform of $g(z)$ at $s=1$:

$\sum_{m=1}^{\infty} \frac{m!}{m^{m}} = \sum_{m=1}^{\infty} \frac{1}{m^{m}} \int_{0}^{\infty} t^{m}e^{-t} dt = \int_{0}^{\infty} e^{-t} \sum_{m=1}^{\infty} \frac{t^{m}}{m^{m}} dt$

# new videos

First, I’ve rendered a 1024×1024 video of the Newton’s Method of the third Jacobi theta function ϑ3(z,q), for |q| = 0.90190165617299947 or thereabouts. This piece is entitled The Swift Luminescent Energy Drink of the Psyche, or When Goorialla Whirls and Whorls and Roars

Secondly, using GAP, I’ve made a video of all Cayley tables of finite groups of order 128:

And finally, A video showing Klein’s j-invariant $j(\tau)$ under the transformation $\tau\rightarrow -1/\tau$ in $\mathbf{SL}(2,\mathbb{R})$: