Half-exponential function

In mathematics, a half-exponential function is a functional square root of an exponential function, that is, a function ƒ that, if composed with itself, results in an exponential function:[1][2]

${\displaystyle f(f(x))=ab^{x}.}$

Another definition is that ƒ is half-exponential if it is non-decreasing and ƒ⁻¹(x^C) ≤ o(log x). for every C > 0.[3]

It has been proven that if a function ƒ is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then ƒ(ƒ(x)) is either subexponential or superexponential.[4][5] Thus, a Hardy L-function cannot be half-exponential.

There are infinitely many functions whose self-composition is the same exponential function as each other. In particular, for every ${\displaystyle A}$ in the open interval ${\displaystyle (0,1)}$ and for every continuous strictly increasing function 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 ${\displaystyle f(f(x))=\exp x}$.[6] The function ${\displaystyle f}$ is the unique solution to the functional equation

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

Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.[2]