Half-exponential function

In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function $$f$$ such that $$f$$ 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
Any exponential function can be written as the self-composition $$f(f(x))$$ for infinitely many possible choices of $$f$$. 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 $$ f (x) = \begin{cases} g (x) & \mbox{if } x \in [0,A], \\ \exp g^{-1} (x) & \mbox{if } x \in (A,1], \\ \exp f ( \ln x) & \mbox{if } x \in (1,\infty), \\ \ln f ( \exp x) & \mbox{if } x \in (-\infty,0). \\ \end{cases} $$

A simple example, which leads to $$f$$ having a continuous first derivative everywhere, is to take $$A=\tfrac12$$ and $$g(x)=x+\tfrac12$$, giving $$ f (x) = \begin{cases} \log_e\left(e^x +\tfrac12\right) & \mbox{if } x \le -\log_e 2, \\ e^x - \tfrac12 & \mbox{if } {-\log_e 2} \le x \le 0, \\ x +\tfrac12 & \mbox{if } 0 \le x \le \tfrac12, \\ e^{x-1/2} & \mbox{if } \tfrac12 \le x \le 1, \\ x \sqrt{e} & \mbox{if } 1 \le x \le \sqrt{e}, \\ e^{x / \sqrt{e}} & \mbox{if } \sqrt{e} \le x \le e, \\ x^{\sqrt{e}} & \mbox{if } e \le x \le e^{\sqrt{e}}, \\ e^{x^{1/\sqrt{e}}} & \mbox{if } e^{\sqrt{e}} \le x \le e^e, \ldots\\ \end{cases} $$

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 $C>0$.