# Elementary function

**Elementary function** is either one or the first three Ackermann functions,
or any finite combination of the first three Ackermann functions functions and their inverse functions.

The first three ackermann functions are

ackermann$_{1,x}(y)=x+y$

ackermann$_{2,x}(y)=x\, y$

ackermann$_{3,x}(y)=\exp_x(y)=x^y~$

## Contents |

## Highest ackermanns

Ackermanns $A_{b,n}$ appear as holomorphic solutions of the transfer equation

$A_{b,n}(z\!+\!1)=A_{b,n-1}\!\big( A_{b,n}(z)\big)$

with additional condition $A_{b,n}(0)=1$, and condition of moderate (slower than exponential) growth in the direction of imaginary axis. For positive $b>0$, the real–holomorphism of ackermanns is assumed, $A_{b,n}(z^*)=A_{b,n}(z)^*$. Each ackermann, except the first one, appears as superfunction of the precious ackermann and Abel function for the next ackermann.

Highest ackermanns $A_{b,n}$, since $n\!>\!3$, are excluded from the set of elementary functions. In particular,

ackermann$_{4,x}(y)=$tet$_x(y)$

ackermann$_{5,x}(y)=$pen$_x(y)$

are not elementary functions. However, some of them are described and should be qualified as special functions.

## Primary elementary functions

First three ackermann and few their combinations, and the inverse functions are called Primary elementary functions. They are defined with the list below:

cos$(x)=\displaystyle \frac{\exp(\mathrm i x) + \exp(-\mathrm i x)}{2}$

sin$(x)\,=\displaystyle\frac{\exp(\mathrm i x) - \exp(-\mathrm i x)}{2\, \mathrm i}$

tan$(x)\,=\displaystyle \frac{\sin(x)}{\cos(x)}$

The inverse functions are also considered as primary elementary functions:

Subtraction$(x,y)=x-y$

Division$(x,y)=x/y$

RootFinding$(x,y)= ~^x\!\sqrt{y}= y^{-x}$

ln$\,=\exp^{-1}$

Arccos$\, =\cos^{-1}$

Arcsin$\, =\sin^{-1}$

Arctan$\,=\tan^{-1}$

## Confusions

In the scientific and educational literature, especially in Physics and Mathematics, the special notations are used, when superscript after the name of a function indicates not number to iterate of this function, but the argument of exponentiation, that should be performed after evaluation of the function; for example,

$\sin^n(x)$ is used in such a way, that it means $\sin(x)^n$.

Such a confusion is often observed namely with elementary functions.

Especially explicit this confusion becomes at $n\!=\!-1$; in the confusive notations,

$\sin^{-1}(x)~$ may mean $~\sin(x)^{-1}\!=\!\frac{1}{\sin(x)}~$ as well as $~\arcsin(x)~$, and only from the context the reader is supposed to guess, which case do the authors mean.

## References

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf

http://mizugadro.mydns.jp/PAPERS/2014acker.pdf

D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

http://www.pphmj.com/references/8246.htm

http://mizugadro.mydns.jp/PAPERS/2014susin.pdf

http://mizugadro.mydns.jp/PAPERS/2014susinL.pdf

D.Kouznetsov. Super sin. Far East Journal of Mathematical Science, v.85, No.2, 2014, pages 219-238.

## Keywords

Ackermann function, Cos, Exp, Logarithm, Sin, Special function, Tan,