Half-exponential function
In mathematics, a half-exponential function is a functional square root of an exponential function, that is, a function ƒ that, if composed with itself, results in an exponential function:
Another definition is that ƒ is half-exponential if it is non-decreasing and ƒ−1 ≤ o.
for every C > 0.
It has been proven that if a function ƒ is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then ƒ is either subexponential or superexponential. 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 in the open interval and for every continuous strictly increasing function g from onto, there is an extension of this function to a continuous monotonic function on the real numbers such that. The function is the unique solution to the functional equation
Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.
