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 ${\displaystyle b}$.

Hellmuth Kneser first proposed a holomorphic construction of the solution of ${\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}$ in 1950. It is closely related to the problem of extending tetration to non-integer values; the value of ${\displaystyle {}^{\frac {1}{2}}a}$ can be understood as the value of ${\displaystyle f{\bigl (}1)}$, where ${\displaystyle f{\bigl (}x)}$ satisfies ${\displaystyle f{\bigl (}f(x){\bigr )}=a^{x}}$. Example values from Kneser's solution of ${\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}$ include ${\displaystyle f{\bigl (}0)\approx 0.49856}$ and ${\displaystyle f{\bigl (}1)\approx 1.64635}$.