# Iterate of exponential

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff) $$u\!+\!\mathrm i v=\exp^1(x\!+\!\mathrm i y)=\exp(x\!+\!\mathrm i y)$$ $$u+\mathrm i v=\exp^{-0.1}(x+\mathrm i y)=\ln^{0.1}(x+\mathrm i y)$$ $$u+\mathrm i v=\exp^{-0.5}(x+\mathrm i y)=\ln^{0.5}(x+\mathrm i y)$$ $$u+\mathrm i v=\exp^{-0.9}(x+\mathrm i y)=\ln^{0.9}(x+\mathrm i y)$$

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 .

## 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 , and the efficient C++ complex double implementation are described in 2010 in Vladikavkaz mathematical Journal in Russian; the English version is also loaded . 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 . 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.