Ultra exponential function

From Wikipedia, the free encyclopedia

 This article or section is in need of attention from an expert on the subject.

WikiProject Mathematics or the Mathematics Portal may be able to help recruit one.

If a more appropriate WikiProject or portal exists, please adjust this template accordingly.
This article or section has multiple issues. Please help improve the article or discuss these issues on the talk page.

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.

Contents

[edit] Definition

Uxpa(x)  : Divergently ultra exponential curve.
Uxpa(x)  : Divergently ultra exponential curve.
Uxpa(x)  : Convergently ultra exponential curve.
Uxpa(x)  : Convergently ultra exponential curve.
Uxpa(x)  : Increasingly Convergently ultra exponential curve.
Uxpa(x)  : Increasingly Convergently ultra exponential curve.
Uxpa(x)  : Unbounded ultra exponential curve.
Uxpa(x)  : Unbounded ultra exponential curve.
Uxpa(x)  : Convex ultra exponential curve.
Uxpa(x)  : Convex ultra exponential curve.
Infinite power tower.
\lim_{n\rightarrow \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\rightarrow \infty} a^{\frac{n}{}} were proved by Leonard Euler.

Hooshmand[1] defined the "ultra exponential function" using the functional equation f(x) = af(x − 1).

A main theorem in Hoooshmand's paper states: Let 0<a\neq 1. If f:(-2,+\infty)\rightarrow \mathbb{R} satisfies the conditions:

  • f(x)=a^{f(x-1)} \; \; \mbox{for all} \; \; x>-1, \; f(0)=1 ,
  • f is differentiable on ( − 1,0),
  • f\; ^' is a nondecreasing or nonincreasing function on ( − 1,0),
  • \lim_{x\rightarrow 0^+}f'(x)=(\ln a) \lim_{x\rightarrow0^-} f'(x),

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 expa.

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

[edit] 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 0 < a < 1, then a^{ \frac{x}{}} can be defined on a larger domain than (-2,+\infty).

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... .

[edit] Natural ultra exponential function

The "natural[dubious – discuss] 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}'(x) is increasing on [-1,+\infty), so \operatorname{uxp}(x) is convex on [-1,+\infty).

The function \chi=\operatorname{uxp}' 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)\rightarrow \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'_-(0)\leq f'_+(0),

then f = uxp.

[edit] Ultra exponential curves

There are five kinds of graph for the ultra exponential functions, depended on range 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 a\geq e .

[edit] Infra logarithm function

If a > 1 , then the ultra exponential function is invertible. Hooshmand denotes its inverse function by Ioga and calls it the "infra logarithm function". The infra logarithm function satisfies the functional equation f(ax) = f(x) + 1.

[edit] See also

[edit] References

  1. ^ M.H. Hooshmand, August 2006, "Ultra power and ultra exponential functions", Integral Transforms and Special Functions, Vol. 17, No. 8, 549-558.