Half-exponential function
In mathematics, a half-exponential function is a functional square root of an exponential function.
composed with itself results in an exponential function:[1][2]
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 the open interval
and for every continuous strictly increasing function
, there is an extension of this function to a continuous strictly increasing function
on the real numbers such that
= exp x
is the unique solution to the functional equation
exp
exp f ( ln x )
ln f ( exp x )
A simple example, which leads to
having a continuous first derivative
increasing, for all real
log
log
log
Crone and Neuendorffer claim that there is no semi-exponential function f(x) that is both (a) analytic and (b) always maps reals to reals.
The piecewise solution above achieves goal (b) but not (a).
Achieving goal (a) is possible by writing
as a Taylor series based at a fixpoint Q (there are an infinitude of such fixpoints, but they all are nonreal complex, for example
), making Q also be a fixpoint of f, that is
, then computing the Maclaurin series coefficients of
Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.
grows at least as quickly as some half-exponential function (its composition with itself grows exponentially) if it is non-decreasing and
