Half-exponential function

In mathematics, a half-exponential function is a function ƒ so that if ƒ is composed with itself the result is exponential:^[1]

$f(f(x)) = ab^x. \,$

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.^[2]

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.^{[3] [4]} 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 $A$ 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 monotonic function $f$ on the real numbers such that $f(f(x))=\exp x$.^[5] The function $f$ is the unique solution to the functional equation

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

Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.^[1]

See also

• Functional square root

References

1. Peter Bro Miltersen, N. V. Vinodchandran, Osamu Watanabe (1999). "Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy". Lecture Notes in Computer Science 1627: 210–220. doi:10.1007/3-540-48686-0_21.
2. Alexander A. Razborov and Steven Rudich (August 1997). "Natural Proofs". Journal of Computer and System Sciences 55 (1): 24–35. doi:10.1006/jcss.1997.1494.
3. http://mathoverflow.net/questions/45477/closed-form-functions-with-half-exponential-growth
4. "Shtetl-Optimized » Blog Archive » My Favorite Growth Rates". Scottaaronson.com. 2007-08-12. Retrieved 2014-05-20.
5. 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 943525.

External links

1. http://mathoverflow.net/questions/12081/does-the-exponential-function-have-a-square-root
2. http://mathoverflow.net/questions/45477/closed-form-functions-with-half-exponential-growth