# 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 [1] 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 [2] 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" [3] that opened the research of various superfunctions. In the similar way, the half-iteration of the logistic sequence [4] is constructed in terms of the superfunction and the Abel function, and similarly, the square root of factorial [5] 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 [6], 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 [7][8][9].

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.

## References

1. http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 H.Kneser. Reelle analytische Lösungen der Gleichung $\varphi(\varphi(x))=e^x$. Equationes Mathematicae, Journal fur die reine und angewandte Mathematik 187 56–67 (1950)
2. http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
3. http://en.wikipedia.org/wiki/Holy_Grail
4. http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
5. http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
6. http://www.springerlink.com/content/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, 81, p.65-76 (2011)
7. http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
8. http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
9. http://www.ils.uec.ac.jp/~dima/PAPERS/2011e1e.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of computation, in press, 2011.