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

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.

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