Half-exponential function

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

Impossibility of a closed-form formula

If a function [math]\displaystyle{ f }[/math] is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then [math]\displaystyle{ f\bigl(f(x)\bigr) }[/math] 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 [math]\displaystyle{ f(f(x)) }[/math] for infinitely many possible choices of [math]\displaystyle{ f }[/math]. In particular, for every [math]\displaystyle{ A }[/math] in the open interval [math]\displaystyle{ (0,1) }[/math] and for every continuous strictly increasing function [math]\displaystyle{ g }[/math] from [math]\displaystyle{ [0,A] }[/math] onto [math]\displaystyle{ [A,1] }[/math], there is an extension of this function to a continuous strictly increasing function [math]\displaystyle{ f }[/math] on the real numbers such that [math]\displaystyle{ f\bigl(f(x)\bigr)=\exp x }[/math]. The function [math]\displaystyle{ f }[/math] is the unique solution to the functional equation [math]\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} }[/math]

Example of a half-exponential function

A simple example, which leads to [math]\displaystyle{ f }[/math] having a continuous first derivative everywhere, is to take [math]\displaystyle{ A=\tfrac12 }[/math] and [math]\displaystyle{ g(x)=x+\tfrac12 }[/math], giving [math]\displaystyle{ 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} }[/math]

Application

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

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