Half-exponential function

In mathematics, a half-exponential function is a functional square root of an exponential function, that is, a function $f$ that, if composed with itself, results in an exponential function: $$f{\bigl (}f(x){\bigr )}=ab^{x},$$ for some constants $a$ and $b$.

Impossibility of a closed form formula

If a function $f$ is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then $f{\bigl (}f(x){\bigr )}$ is either subexponential or superexponential. Thus, a Hardy L-function cannot be half-exponential.

Construction

There are infinitely many functions whose self-composition is the same exponential function as each other. In particular, for every $A$ in the open interval $(0,1)$ and for every continuous strictly increasing function $g$ from $[0,A]$ onto $[A,1]$, there is an extension of this function to a continuous strictly increasing function $f$ on the real numbers such that $f{\bigl (}f(x){\bigr )}=\exp x$. The function $f$ is the unique solution to the functional equation $$f(x)={\begin{cases}g(x)&{\mbox{if }}x\in [0,A],\\\exp g^{-1}(x)&{\mbox{if }}x\in (A,1],\\\exp f(\ln x)&{\mbox{if }}x\in (1,\infty ),\\\ln f(\exp x)&{\mbox{if }}x\in (-\infty ,0).\\\end{cases}}$$

Example of a half-exponential function

A simple example, which leads to $f$ having a continuous first derivative everywhere, is to take $A={\tfrac {1}{2}}$ and $g(x)=x+{\tfrac {1}{2}}$, giving $$f(x)={\begin{cases}\log _{e}\left(e^{x}+{\tfrac {1}{2}}\right)&{\mbox{if }}x\leq -\log _{e}2,\\e^{x}-{\tfrac {1}{2}}&{\mbox{if }}-\log _{e}2\leq x\leq 0,\\x+{\tfrac {1}{2}}&{\mbox{if }}0\leq x\leq {\tfrac {1}{2}},\\e^{x-1/2}&{\mbox{if }}{\tfrac {1}{2}}\leq x\leq 1,\\x{\sqrt {e}}&{\mbox{if }}1\leq x\leq {\sqrt {e}},\\e^{x/{\sqrt {e}}}&{\mbox{if }}{\sqrt {e}}\leq x\leq e,\\x^{\sqrt {e}}&{\mbox{if }}e\leq x\leq e^{\sqrt {e}},\\e^{x^{1/{\sqrt {e}}}}&{\mbox{if }}e^{\sqrt {e}}\leq x\leq e^{e},\ldots \\\end{cases}}$$

Application

Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential. A function $f$ grows at least as quickly as some half-exponential function (its composition with itself grows exponentially) if it is non-decreasing and $f^{-1}(x^{C})=o(\log x)$, for every $C>0$.

See also

• Iterated function – Result of repeatedly applying a mathematical function
• Schröder's equation – Equation for fixed point of functional composition
• Abel equation – Equation for function that computes iterated values

External links

• Does the exponential function have a (compositional) square root?
• “Closed-form” functions with half-exponential growth