Half-exponential function

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

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 ${\displaystyle g}$ 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

1. Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung φ(φ(x) = e^x und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik . 187: 56–67. MR   0035385.
2. Miltersen, Peter Bro; Vinodchandran, N. V.; Watanabe, Osamu (1999). "Super-polynomial versus half-exponential circuit size in the exponential hierarchy". In Asano, Takao; Imai, Hiroshi; Lee, D. T.; Nakano, Shin{-}Ichi; Tokuyama, Takeshi (eds.). Computing and Combinatorics, 5th Annual International Conference, COCOON '99, Tokyo, Japan, July 26–28, 1999, Proceedings. Lecture Notes in Computer Science. Vol. 1627. Springer. pp. 210–220. doi:10.1007/3-540-48686-0_21. MR   1730337.
3. van der Hoeven, J. (2006). Transseries and real differential algebra. Lecture Notes in Mathematics. Vol. 1888. Springer-Verlag, Berlin. doi:10.1007/3-540-35590-1. ISBN   978-3-540-35590-8. MR   2262194.. See exercise 4.10, p. 91, according to which every such function has a comparable growth rate to an exponential or logarithmic function iterated an integer number of times, rather than the half-integer that would be required for a half-exponential function.
4. Crone, Lawrence J.; Neuendorffer, Arthur C. (1988). "Functional powers near a fixed point". Journal of Mathematical Analysis and Applications. 132 (2): 520–529. doi: 10.1016/0022-247X(88)90080-7 . MR   0943525.
5. Razborov, Alexander A.; Rudich, Steven (1997). "Natural proofs". Journal of Computer and System Sciences . 55 (1): 24–35. doi: 10.1006/jcss.1997.1494 . MR   1473047.

External links

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