Ultra exponential function

Articleissues
OR=January 2008
other=This article uses nonstandard notations, which are confusing and superfluous.In mathematics the ultra exponential function is a special case of the iterated exponential function more commonly known as tetration, with specific extension to non-integer values of the argument.

Definition

thumb|150px|">$Uxp\_a(x)$ : Divergently ultra exponential curve.
thumb|150px|">$Uxp\_a(x)$ : Convergently ultra exponential curve.
thumb|150px|">$Uxp\_a(x)$ : Increasingly Convergently ultra exponential curve.
thumb|150px|">$Uxp\_a(x)$ : Unbounded ultra exponential curve.
thumb|150px|">$Uxp\_a(x)$ : Convex ultra exponential curve.
thumb|150px|">$lim\_\{n\; ightarrow\; infty\}\; x^\{frac\{n\}\{$ Infinite power tower.Let "a" be a positive real number. The notation $a^\{frac\{n\}\{$which is defined by $a^\{frac\{n\}\{\; =a^\{a^\{frac\{n-1\}\{\}$ ($n=2,3,4,cdots$), $a^\{frac\{1\}\{=\; a$ is called "a" to the "ultra power" of "n". In other words $a^\{frac\{n\}\{=a^\{a^\{.^\{.^\{.^a$, "n" times. For other notations see tetration.

Necessary and sufficient conditions for the convergence of $lim\_\{n\; ightarrow\; infty\}\; a^\{frac\{n\}\{$ were proved by Leonard Euler.Hooshmandcite journal
author=M.H.Hooshmand,
year=2006
title=Ultra power and ultra exponential functions
journal=Integral Transforms and Special Functions
volume=17
issue=8
pages=549-558
doi= 10.1080/10652460500422247
url=http://www.informaworld.com/smpp/content~content=a747844256?words=ultra%7cpower%7cultra%7cexponential%7cfunctions&hash=721628008 ] defined the "ultra exponential function" using the functional equation $f(x)=a^\{f(x-1)\}$. A main theorem in Hoooshmand's paper states: Let

*$f(x)=a^\{f(x-1)\}\; ;\; ;\; mbox\{for\; all\}\; ;\; ;\; x>-1,\; ;\; f(0)=1$ ,
*$f$ is differentiable on $(-1,0)$,
* $f\text{'}$ is a nondecreasing or nonincreasing function on $(-1,0),$
*$f\text{'}left(0^+\; ight)=(ln\; a)\; f\text{'}left(0^-\; ight),$ or $f\text{'}left(-1^+\; ight)\; =\; f\text{'}left(0^-\; ight).$

then $f$ is uniquely determined through the equation

:$;\; ;\; ;\; f(x)=exp^\{\; [x]\; \}\_a\; (a^\{(x)\})=exp^\{\; [x+1]\; \}\_a((x))\; ;\; ;\; ;\; mbox\{for\; all\}\; ;\; ;\; x>-2$,

where $(x)=x-\; [x]$ denotes the fractional part of x and $exp^\{\; [x]\; \}\_a$ is the $[x]$-iterated function of the function $exp\_a$.

The ultra exponential function is then defined as $mbox\{uxp\}\_a(x)=exp^\{\; [x+1]\; \}\_a((x))\; ;\; ;\; ;\; mbox\{for\; all\}\; ;\; ;\; x>-2$.

Ultra power

Since $mbox\{uxp\}\_a(n)=a^\{\; frac\{n\}\{$, for every positive integer $n$, and because of the uniqueness theorem, the definition of ultra power is extended by $a^\{\; frac\{x\}\{=\; mbox\{uxp\}\_a(x)$ . If

Examples:$a^\{frac\{0\}\{=1,;\; a^\{frac\{-1\}\{=0,;\; 2\{\; frac\{4\}\{=65536,;\; e\{\; frac\{pi\; /2\}\{=5.868...,;\; 0.5^\{\; frac\{-4.3\}\{=4.03335...,;$ $0.6^\{\; frac\{-5.264\}\{=-5.35997...,$$0.7^\{\; frac\{3.1\}\{=0.7580...$ .

Natural ultra exponential function

The "naturalDubious|date=March 2008 ultra exponential function" $e^\{frac\{x\}\{$, denoted by $operatorname\{uxp\}(x)$, is continuously differentiable, but its second derivative does not exist at integer values of its argument.

$operatorname\{uxp\}\text{'}(x)$ is increasing on $[-1,+infty)$, so $operatorname\{uxp\}(x)$ is convex on $[-1,+infty)$.

The function $chi=operatorname\{uxp\}\text{'}$ satisfies the following functional equation (difference equation):

:$;\; ;\; ;\; ;\; chi(x)=e^\{\; frac\{x\}\{\; chi(x-1)\; ;\; ;\; ;\; mbox\{for\; all\}\; ;\; ;\; x>-1.$

There is another uniqueness theorem for the natural ultra exponential function that states: If $f:\; (-2,\; +infty)\; ightarrow\; mathbb\{R\}$ is a function for which:
* $f(x)=e^\{f(x-1)\}\; ;\; ;\; ;\; mbox\{for\; all\}\; ;\; ;\; x>-1,$
* $f(0)=1,$
* $f$ is convex on $(-1,0)$,
* $f\text{'}\_-(0)leq\; f\text{'}\_+(0),$ then $f=mbox\{uxp\}.$

Ultra exponential curves

There are five kinds of graph for the ultra exponential functions, depended onrange values of $a$ (figures 1-5). If $a>e^\{frac\{1\}\{e$, then the ultra exponential curve is upper and lower unbounded. It is convex from a number on, if $ageq\; e$.

Infra logarithm function

If $a>1$ , then the ultra exponential function is invertible. Hooshmand cite journal
author=M.H.Hooshmand,
year=2008
title=Infra logarithm and ultra power part functions
journal=Integral Transforms and Special Functions
volume=19
issue=7
pages=497-507
doi= 10.1080/10652460801965555
url=http://www.informaworld.com/smpp/content~content=a901619070~db=all?logout=true ] denotes its inverse function by $Iog\_a$ and calls it the "infra logarithm function". The infra logarithm function satisfies the functional equation $f(a^x)=f(x)+1$.

ee also

* Tetration
* Iterated function
* Iterated logarithm

References