In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function such that composed with itself results in an exponential function:[1][2]
If a function is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then is either subexponential or superexponential.[3] Thus, a Hardy L-function cannot be half-exponential.
Any exponential function can be written as the self-composition for infinitely many possible choices of . In particular, for every in the open interval and for every continuous strictly increasing function from onto , there is an extension of this function to a continuous strictly increasing function on the real numbers such that .[4] The function is the unique solution to the functional equation
A simple example, which leads to having a continuous first derivative everywhere, is to take and , giving
Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.[2] A function grows at least as quickly as some half-exponential function (its composition with itself grows exponentially) if it is non-decreasing and , for every .[5]