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]

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

for some constants ${\displaystyle a}$ 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

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

Example of a half-exponential function

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]

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

References

External links

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