Elementary function

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



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,



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:



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


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

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


Nonholomorphic functions

Often, the set of primary elementary function is extended with four non-holomorphic functions, namely,

Re$(z)\!=\!\Re(z)$ that returns the real part of the argument,

Im$(z)\!=\!\Im(z)$ that returns the Imaginary part of the argument,

Abs$(z) \!=\! |z| \!=\! $ $\sqrt{\Re(z)^2+\Im(z)^2}$, that is returns the absolute value of the argument

Arg$(z)\!=$Arctan$\big(\Im(z),\Re(z)\big)$, that returns the phase of the argument.

The last function sometimes is called "argument", but such notation may cause confusion with object, that appears in parenthesis after the name of the function, and also qualified as "argument". For this reason, in TORI, function Arg is called phase, not "argument". However, this also may cause confusions, for example, at the attempts to apply such an interpretation to the melting, as the phase transition of a chemical from the solid phase to the liquid phase.

In many programming languages (except Mathematica), Arctan of two arguments is denoted as atan2. Usually, the compilers count number of arguments after the name of a function, and distinguish a function of single argument from that of two arguments.

For $\Re(z)\!>\!0$, the the phase can be expressed also through arctangent of single argument: Arg$(z)\!=$Arctan$\big(\Im(z)/\Re(z)\big)$

Then, any combination of the primary elementary functions with first 3 ackermanns and their inverse functions also van be qualified as elementary function.

In order to avoid confusions in the mathematical deduction, meaning of term elementary function should be confirmed in any physical and mathematical text.


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.

Special function

Elementary function appears as "special case" of a special function.

Any function is qualified as "special", if the properties of this function are known and at least one efficient algorithm of the evaluation is supplied.

Practically, the special functions can be evaluated with complex double precision in a real time; the case To press a key, to have a tea barely can be qualified as "special function".

The elementary functions and, especially, the primary elementary functions are extremely fast to evaluate. However, the complicated combination of primary elementary functions may be a little bit slow at the evaluation; sometimes, the straightforward implementation may cause loss of precision.

Usually, any representation of solution through special function, and, in particular, through the elementary function, is qualified as exact solution.


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

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


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