# Iterate of exponential

$y=\exp^n(x)~$ versus real $x$ for various $n$
$u\!+\!\mathrm i v=\exp^1(x\!+\!\mathrm i y)=\exp(x\!+\!\mathrm i y)$
$u+\mathrm i v=\exp^{0.9}(x+\mathrm i y)$
$u+\mathrm i v=\exp^{0.5}(x+\mathrm i y)$
$u+\mathrm i v=\exp^{0.1}(x+\mathrm i y)$
$u+\mathrm i v=\exp^{-0.1}(x+\mathrm i y)=\ln^{0.1}(x+\mathrm i y)$
$u+\mathrm i v=\exp^{-0.5}(x+\mathrm i y)=\ln^{0.5}(x+\mathrm i y)$
$u+\mathrm i v=\exp^{-0.9}(x+\mathrm i y)=\ln^{0.9}(x+\mathrm i y)$
$u+\mathrm i v=\exp^{-1}(x+\mathrm i y)=\ln(x+\mathrm i y)$

Iteration of exponential (or Iteration of exponent) is function $f(z)=\exp^n(z)$, where upper superscript indicates the number of iteration.

In TORI, the number in superscript after a name of any function denotes the number of iteration. This notation is neither new, nor original; Walter Bergweiler had used it in century 20 [1].

## Integer and non-integer $n$

The most often are the first iteration of exponent, $n=1$; $\exp^1=\exp$

and the minus first iteration, $n=-1$; $\exp^{-1} = \ln$.

Less often they appear with $n = \pm 2$; $\exp^2(z)=\exp(\exp(z))$, and $\exp^{-2}(z)=\ln(\ln(z))$. Other values of number of iteration are not usual, and until year 2008, there was no regular way to evaluate iteration of exponential for any non–integer number $n$ of iteration. However, with tetration tet, that is superfunction of exponent, and Arctetration ate, that is Abel function of exponent, the $n$th iteration can be expressed as follows:

$\exp^n(z)=\mathrm{tet}(n+\mathrm{ate}(z))$

Both, tet and ate are holomorphic functions; so, the representation above can be used for non-integer $n$. The exponential can be iterated even complex number of times.

## Iimplementation

Representation of $\exp^n$ through function tet and ate defines the $n$th iterate of exponential for any complex number $n$ of iterations. Methods for the evaluation are described in 2009 by D.Kouznetsov in Mathematics of Computation [2], and the efficient C++ complex double implementation are described in 2010 in Vladikavkaz mathematical Journal in Russian; the English version is also loaded [3]. WIth known properties and the efficient implementation, functions tet, ate and non–integer ietrations of the exponent shouls be qualified as special functions; in computation, one can access them as if they would be elementary functions. The complex doube implementations of functions tet and ate are loaded to TORI, see fsexp.cin and fslog.cin; they run at various operational systems; at least under Linux and Macintosh. Reports of any problems with the use or the reproducible bugs should be appreciated.

Complex maps of the $n$th iteration of exponential, $f=\exp^n(x+\mathrm i y)$ are shown in figures at right with lines $u=\Re(f)$ and lines $v=\Im(f)$ for various values $n$ in the $x$,$y$ plane. As the function is real-holomorphic, the maps are symmetric; so the only upper half plane is shown in the figures.

## Cut lines

While $n$ is not integer, $\exp^n(z)$ is holomorphic in the complex plane with two cut lines $\Re(z)\le \Re(L)$, $\Im(z)=\pm \Im(L)$, where $L\approx 0.3+1.3 \mathrm i$ is fixed point of logarithm, id est, solution of equation

$L=\ln(L)$.

In the figures at right, one of these cuts is seen; it is marked with dashed line. The additional levels $\Re(L)$ for the real part of $\exp^n$ and $\Im(L)$ for the imaginary part are drown with thick green lines; of course, these lines cross each other at the branch point $L$.

In addition, for negative number $n$ of iterations (and, in particular, for $n=-1$), there is cut line along the negative part of the real axis, id est, from $-\infty$ to $\mathrm{tet}(-2-n)$.

## Special function

Properties of the iteration of the exponential are described.

$\exp^n(z)$ is holomorphic function with respect to $z$ and with respect to $n$. Properties of this function are analyzed and described. The efficient (fast and precise) algorithm for the evaluation is supplied with routines fsexp.cin and cslog.cin.

With achievements above, function $(n,z) \mapsto \exp^n(z)$ is qualified as special function. Designers of compilers and interpreters from the programming languages are invited to borrow the implementations of tetration and arctetration in order to provide the built-in function, that evaluates $\exp^n(z)$ for complex values of $n$ and $z$. In particular, in Mathematica, there is already name for such a function; it should be called with Nest[Exp,z,n]. Up to year 2013, the built-in function Nest is implemented in such a way, that number $n$ of iteration should be expressed with natural constant, positive integer number [4]. Over-vice, the built-in function generates the error message instead of to perform the calculations and evaluations requested. With use of superfunctions and Abel functions, Nest could be implemented for more general case.

## References

1. http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188.
2. http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D. Kouznetsov. Solution of $F(x+1)=\exp(F(x))$ in complex $z$-plane. 78, (2009), 1647-1670