File:Superfactorea500.png

Real-real plot of

$y\!=\!\mathrm{Factorial}(x) ~ $ , blue curve,

$y\!=\!\mathrm{SuperFactorial}(x)\!=\!\mathrm{Factorial}^x(3) ~ $ , red curve,

versus $x$

Copyleft 2011 by Dmitrii Kouznetsov.

Factorial

Factorial is meromorphic function;

$\mathrm{Factorial}(z)=z \times \mathrm{Factorial}(z\!-\!1) ~ \forall z\in \mathbb C \backslash \{ -n, n\in \mathbb N \}$

$ \mathrm{Factorial}(z^*)=\mathrm{Factorial}(z)^*$

$ \displaystyle \lim_{x\rightarrow -\infty} \mathrm{Factorial}(x\!+\! \mathrm i y)=0

~ \forall y\in \mathbb R : y\ne 0$

SuperFactorial

SuperFactorial is superfunction of Factorial constructed with regular iteration at its fixed point 2;

$ \mathrm{SuperFactorial}(z^*)=\mathrm{SuperFactorial}(z)^*$

$ \mathrm{SuperFactorial}(z)=\mathrm{Factorial}^x(3)$

$\displaystyle \lim_{x\rightarrow -\infty} \mathrm{SuperFactorial}(x\!+\! \mathrm i y)=2 ~ \forall y\in \mathbb R$

$ \mathrm{SuperFactorial}(3)=0$

In the first description ^[1] of SuperFactorial, its value at zero (last condition above) is not adjusted.

Generators

This image is generated with the following sources:

1. fac.cin , the complex double implementation of factorial
2. SuperFactorial.cin , the complex double implementation of superfactorial
3. ado.cin , that writes the header of the EPS file
4. Superfactoreal.cc , that plots the curves as superfactoreal.pdf
5. Superfactorea.tex , that add labels, making new PDF

After the generation, the output file superfactorea.pdf is converted to superfactorea500.png using the resoluton "500".

The generators of the figure are misplaced (or misnamed) and cannot be loaded here. Therefore, the similar figure with generators is loaded: http://tori.ils.uec.ac.jp/TORI/index.php/File:SuperFacPlotT.png

I did not prepare the special implementation for the real values of the argument; so, for the real plots, the real part of the output is used.

References