Английская Википедия:Half-exponential function
In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function <math>f</math> such that <math>f</math> composed with itself results in an exponential function:Шаблон:R <math display=block>f\bigl(f(x)\bigr) = ab^x,</math> for some constants Шаблон:Nowrap
Impossibility of a closed-form formula
If a function <math>f</math> is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then <math>f\bigl(f(x)\bigr)</math> is either subexponential or superexponential.Шаблон:R Thus, a [[Hardy field#Examples|Hardy Шаблон:Mvar-function]] cannot be half-exponential.
Construction
Any exponential function can be written as the self-composition <math>f(f(x))</math> for infinitely many possible choices of <math>f</math>. In particular, for every <math>A</math> in the open interval <math>(0,1)</math> and for every continuous strictly increasing function <math>g</math> from <math>[0,A]</math> onto <math>[A,1]</math>, there is an extension of this function to a continuous strictly increasing function <math>f</math> on the real numbers such that Шаблон:Nowrap The function <math>f</math> is the unique solution to the functional equation <math display=block> 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>
A simple example, which leads to <math>f</math> having a continuous first derivative everywhere, is to take <math>A=\tfrac12</math> and <math>g(x)=x+\tfrac12</math>, giving <math display=block> 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.Шаблон:R A function <math>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>f^{-1}(x^C)=o(\log x)</math>, for Шаблон:Nowrap
See also
References
External links
- Does the exponential function have a (compositional) square root?
- “Closed-form” functions with half-exponential growth
