In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function ${\displaystyle f}$ such that ${\displaystyle f}$ composed with itself results in an exponential function:^[1][2]

$${\displaystyle f{\bigl (}f(x){\bigr )}=ab^{x},}$$

${\displaystyle a}$ and ${\displaystyle b}$.

Impossibility of a closed-form formula

If a function ${\displaystyle f}$ is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then ${\displaystyle f{\bigl (}f(x){\bigr )))$ is either subexponential or superexponential.^[3] Thus, a Hardy L-function cannot be half-exponential.

Construction

Any exponential function can be written as the self-composition ${\displaystyle f(f(x))}$ for infinitely many possible choices of ${\displaystyle f}$. In particular, for every ${\displaystyle A}$ in the open interval ${\displaystyle (0,1)}$ and for every continuous strictly increasing function ${\displaystyle g}$ from ${\displaystyle [0,A]}$ onto ${\displaystyle [A,1]}$, there is an extension of this function to a continuous strictly increasing function ${\displaystyle f}$ on the real numbers such that ${\displaystyle f{\bigl (}f(x){\bigr )}=\exp x}$.^[4] The function ${\displaystyle f}$ is the unique solution to the functional equation

$${\displaystyle f(x)={\begin{cases}g(x)&{\mbox{if ))x\in [0,A],\\\exp g^{-1}(x)&{\mbox{if ))x\in (A,1],\\\exp f(\ln x)&{\mbox{if ))x\in (1,\infty ),\\\ln f(\exp x)&{\mbox{if ))x\in (-\infty ,0).\\\end{cases))}$$

A simple example, which leads to ${\displaystyle f}$ having a continuous first derivative everywhere, is to take ${\displaystyle A={\tfrac {1}{2))}$ and ${\displaystyle g(x)=x+{\tfrac {1}{2))}$, giving

$${\displaystyle f(x)={\begin{cases}\log _{e}\left(e^{x}+{\tfrac {1}{2))\right)&{\mbox{if ))x\leq -\log _{e}2,\\e^{x}-{\tfrac {1}{2))&{\mbox{if )){-\log _{e}2}\leq x\leq 0,\\x+{\tfrac {1}{2))&{\mbox{if ))0\leq x\leq {\tfrac {1}{2)),\\e^{x-1/2}&{\mbox{if )){\tfrac {1}{2))\leq x\leq 1,\\x{\sqrt {e))&{\mbox{if ))1\leq x\leq {\sqrt {e)),\\e^{x/{\sqrt {e))}&{\mbox{if )){\sqrt {e))\leq x\leq e,\\x^{\sqrt {e))&{\mbox{if ))e\leq x\leq e^{\sqrt {e)),\\e^{x^{1/{\sqrt {e))))&{\mbox{if ))e^{\sqrt {e))\leq x\leq e^{e},\ldots \\\end{cases))}$$

Application

Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.^[2] A function ${\displaystyle f}$ grows at least as quickly as some half-exponential function (its composition with itself grows exponentially) if it is non-decreasing and ${\displaystyle f^{-1}(x^{C})=o(\log x)}$, for every ${\displaystyle C>0}$.^[5]