Square root of exponential

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Square root of exponential $$\varphi=\sqrt{\exp}=\exp^{1/2}$$ is half-iteration of the exponential, id est, such function that its second iteration gives the exponential:

$$\!\!\!\!\!\!\!\!\!\!\!\!\!(1) ~ ~ ~ \varphi(\varphi(z))=z$$

Such a $$\varphi$$ is assumed to be holomorphic function for some domain of values of $$z$$.

Function $$\sqrt{\exp}$$ should not be confused with $$z\mapsto \exp(z)^{1/2}$$; in the range of holomorphism, the last can be reduced to $$\exp(z/2)$$.

History

There exist many solutions of equation (1). Some of them are considered in y.1950 by Helmuth Kneser  in the middle or 20 century. But the real-holomorphic solution was not constructed that time.

In the beginning of century 21, the real-holomorphic solution $$\varphi$$ has been suggested  in terms of tetration $$\mathrm{tet}$$ and ArcTetration $$\mathrm{ate}$$,

$$\!\!\!\!\!\!\!\!\!\!\!\!\!(2) ~ ~ ~ \displaystyle \varphi(z)=\sqrt{\exp}(z) = \exp^{1/2}(z)=\mathrm{tet}\Big(\frac{1}{2}+\mathrm{ate}(z)\Big)$$

and namely this function is considered as default square foot of exponential. Expression (2) is just special case of the general expression of the $$c$$th iteration of some transfer function $$T$$ through its superfunction $$F$$ and the corresponding Abel function $$G=F^{-1}$$:

$$\!\!\!\!\!\!\!\!\!\!\!\!\!(3) ~ ~ ~ \displaystyle T^c(z)=F(c+G(z))$$

In the equation (3), the number $$c$$ of iteration can no need to be integer; it can be fractional, irrational or even complex. However, $$T^c(z)$$ should not be confused with $$T(z)^c$$. Equation (2) comes from (3) at $$T\!=\!\exp$$, $$F\!=\!\mathrm{tet}$$ and $$G\!=\!\mathrm{ate}$$.

The square root of exponential seems to be the first non-trivial function for which the non-trivial non-integer iterations were reported, a "Holy Graal cup"  that opened the research of various superfunctions. In the similar way, the half-iteration of the logistic sequence  is constructed in terms of the superfunction and the Abel function, and similarly, the square root of factorial  can be expressed in therms of the SuperFactorial and AbelFactorial (or "ArcSuperFactorial") functions.

Uniqueness

The additional conditions on the behavior of the ArcTetration in vicinity of the fixed points of the logrithm provide the uniqueness of the ArcTetration , and, therefore, the uniqueness of the suggested real-holomorphic square root of the exponential.

Generalization

Equation (2) defines the square root of exponential as the $$\frac{1}{2}$$th iteration of the exponentis. It is special case of equation (3).

The similar expression for the evaluation of the half-iteration can be used also for the exponentials to various values of base, $$\exp_b$$ for $$b\!>\!1$$, as soon as the corresponding tetration and arctetration are implemented .

Square root of various functions can be interpreted as a halfiteration. However, the writing $$T^c(z)$$ should not be confused with $$T(z)^c$$, nor with $$T(z^c)$$; these are pretty different expressions.