Nemtsov function

\(y=\mathrm{nem}_q(x)\)

Complex maps: \(u\!+\!\mathrm i v=f(x\!+\!\mathrm i y)\) for \(f\!=\!\mathrm{nem}_0\), \(f\!=\!\mathrm{ArqNem}_0 ~~\) (top) and for \(f\!=\!\mathrm{nem}_2\), \(f\!=\!\mathrm{ArqNem}_2\) (Bottom)

The Nemtsov function is a special kind of polynomial, suggested as an example of a transfer function in the book «Superfunctions» , 2020.
The description is loaded also as the Mizugadro Preprint .

The Nemtsov function \(y=\mathrm{nem}_q(x)= x+x^3+q x^4\) is shown in figure at right versus \(x\) for various \(q\!\ge\!0\).

Complex maps of \( \mathrm{nem}_q \) are shown in the left-hand column of the figure below.

The right-hand column shows similar maps for the inverse function \( \mathrm{ArqNem}_q = \mathrm{nem}_q^{-1} \).

This article describes the Nemtsov function nem and related functions:
the inverse function ArqNem,
the superfunction SuNem and
the abelfunction AuNem.

Motivation and naming

The Nemtsov function serves as a simple example of a real-holomorphic transfer function with a real fixed point, where the regular iteration method for constructing a superfunction cannot be applied "as is".
The editor did not find any simpler example of this kind, other than the special polynomial of 4th order.

The Nemtsov function is an attempt to propose an "exotic" transfer function for which the superfunction cannot be constructed in the same way as for other superfunctions from the «Table of superfunctions».

The expansion of the Nemtsov function at its fixed point begins with the linear term with coefficient unity. This makes impossible the Regular iteration . (That works well for the exponential to base \(\sqrt{2}\) as the transfer function).

This case is similar to that of the exponential to base \(\exp(1/\mathrm e)\) , but has no quadratic term, so, the exotic iteration cannot be applied "as is".

The case of sin as the transfer function is closer. However, sin is antisymmetric function, \(\sin(-z)=-\sin(z)\). The symmetry simplifies the consideration.
The Nemtsov function \( \mathrm{nem}_q\) at \(q>0\) does not have such a symmetry.
This caused doubts, if the superfunction for the Nemtsov function can be constructed.

In such a way, the Nemtsov function appears as an attempt to built-up the growing real-holomorphic function, for which the growing real-holomorphic superfunction cannot be constructed.

The attempt failed; the Superfunction for the Nemtsov function is constructed and described below.

B.Nemtsov

The function name was introduced in 2015 as a personal memorial reference to Boris Nemtsov. The naming is unrelated to the mathematical properties of the function.
The need of the special name for this function had been revealed 2015.02.27, in the same day when Putin killed Nemtsov. (To year 2025, no other scientific concept of that sad event is found.) Apparently, the total corruption in Russia does not allow the professionals to investigate that case, and that terroristic act caused many publications. This makes the family name "Nemtsov" to be a memorial mark, label on the timeline of Human History.
To year 2025, no better notation for this special polynomial is found. So, term «Nemtsov function» is used here.

The function is denoted \( \mathrm{nem}_q \) (lowercase), while “Nem” in “Nemtsov function” is a proper name and not the function symbol. Following tradition to write names of elementary functions with lowercase letters, in formulas it appears as "nem". However, it is capitalized when it is necessary to separate it from the prefix than may indicate the inverse function ArqNem, the abelfunction AuNem or the superfunction SuNem.

Definition and notations

Let \(q\) be a non-negative real parameter.

Then, Nemtsov function \(\mathrm{nem}\) is defined for complex argument \(z\) as follows:

(1)\(~ ~ ~ ~ ~ ~ \mathrm{nem}_q(z)= z+z^3+q z^4\)

For Function \(~ \mathrm{nem}_q~\) by equation (1), at \(q\!>\!0\), the algorithms, described in the first edition of the Russian version of the book «Суперфункции» (2014) cannot be applied "as is", but a small modification, generalization is required.

For this reason, function \(\mathrm{nem}_q\) by equation (1) is considered as a transfer function.

To year 2025, the Nemtsov function appears to be the last attempt to suggest a real-holomorphic growing transfer function such that its superfunction cannot be constructed with methods similar to those already described in publications.

This attempt fails. For the Nemtsov function, the superfunction SuNem and the abelfunction AuNem are constructed in almost the same method used for the transfer function sin.

In this article, properties of the Nemtsov function are described, and also some properties of the related functions:

Inverse function, denoted with ArqNem,

\(\mathrm{nem}_q(\mathrm{ArqNem}_q(z))=z\)

Superfunction, denoted with SuNem,

\(\mathrm{SuNem}_q(z\!+\!1)=\mathrm{nem}_q\big(\mathrm{SuNem}_q(z)\big)\)

Abel function, denoted with AuNem,

\(\mathrm{AuNem}_q(\mathrm{nem}_q(z))=\mathrm{AuNem}_q(z)+1\)

and the corresponding iterates

\(\mathrm{nem}_q^n(z)=\mathrm{SuNem}_q\big(n+ \mathrm{SuNem}_q(z)\big)\)

The inverse function is called ArqNem. This name allows to distinguish it from other inverse functions of the Nemtsov function. Two other inverse functions are called ArcNem and ArkNem. A priori, it had been difficult to guess, that namely ArqNem happens to be suitable for construction of the superfunction and the corresponding abelfunction; so, all the three versions had been assigned (designated) the different names. These inverse functions have different positions of the cuts of the range of holomorphism.

The Abel function for the Nemtov function is called AuNem, to indicate, that it is constructed by the exotic iterates at the fixed point zero, that is maximal (Upper) among the fixed points of the Nemtsov function.

While no other iterates of the Nemtsov function are presented, no special mark is used to denote the iterates \(\mathrm{nem}_q^n\). Later, perhaps, one additional subscript \(_{\mathrm u}\) will be added to the notation, in order to distinguish this iterate from other iterates, constructed, for example, using the asymptotic behaviour of the Nemtsov function (and its iterates) at infinity.

Inverse function

   

\(x\!+\!\mathrm i y= \mathrm{NemBra}(q) ~\) and \(~ x\!+\!\mathrm i y= \mathrm{NemBran}(q)\)

To construct the inverse function in the complex plane, we need to identify the saddle points and choose the appropriate cut lines. For function ArqNem\(_q\), for \(q=0\) and for \(q=2\), in the maps at the right hand side column of the second figure, these cuts are shown with yellow lines.
These lines connect the saddle points of the Nemtsov function.
Positions of these branch points are considered below.

For real \(q\), the Nemtsov function is real holomorphic in the whole complex plane;

\(\mathrm{nem}_q(z^*)=\mathrm{nem}_q(z)^*\)

At least for positive values of the argument, the Nemtsov function grows monotonically.
The monotonous growth is also property of the inverse function ArqNem, that of the Abelfunction AuNem and that of the real iterates of the Nemtsov function.

The inverse function \(\mathrm{ArqNem}_q\) has the branch points.
Two of them are complex; and one of them is expressed with function NemBran.
Parametric plot of function NemBran is shown at second picture in figure at right.
At \(z=\mathrm{NemBran}(q)\), function \(\mathrm{ArqNem}_q(z)\) has infinite derivative.

For construction of the inverse function, important are the complex solutions \(A\) of equation \(\mathrm{nem}_q^{\prime}(A)=0\).
One of these solutions is expressed with function NemBra, id est, \(A=\mathrm{NemBra}(q)\). Parametric plots of functions NemBra and NemBran are shown at right;

\(\mathrm{NemBran}(q)=\mathrm{nem}_q\big(\mathrm{NemBra}(q)\big)\) :

\( \left( \begin{array}{cc} x=\mathrm{Re}\big(\mathrm{NemBra}(q)\big) \\ y=\mathrm{Im}\big(\mathrm{NemBra}(q)\big) \end{array}\right) ~\), left and \(~ \left( \begin{array}{cc} x=\mathrm{Re}\big(\mathrm{NemBran}(q)\big) \\ y=\mathrm{Im}\big(\mathrm{NemBran}(q)\big) \end{array}\right) ~\) , right.

For positive \(q\), both real and imaginary parts of the branch point are significantly smaller than unity.

Once the function NemBra is implemented, the efficient algorithm for evaluation of the inverse functions of the Nemtsov function can be constructed.
Three of them are denoted with symbols «ArcNem», «ArkNem», «ArqNem». The last one happens to be appropriate for construction of non-integer iterates, holomorphic at least in some vicinity of the positive part of the real axis.

Complex map of function ArqNem are shown in the Second figure.

The C++ complex double implementation of function ArqNem is loaded as arqnem.cin.
In the implementation, parameter \(q\) is stored in the global variable \(Q\). Perhaps, it would not be a good solution for any software, but this happens to be a simple way to plot pictures for book «Superfunctions» .

Before evaluation of \(\mathrm{ArqNem}_q\) of complex argument, the complex branch point should be evaluated with routine nembran.cin and stored in the global variables tr and ti; in the version from year 2015, the real and imaginary parts of the branch point are stored as two global variables. At any change of parameter \(q\), these values should be recalculated.

Superfunction

\(y\!=\!\mathrm{SuNem}_{q}(x)\) for various \(q\)

  \(u\!+\!\mathrm i v=\mathrm{SuNem}_0(x\!+\!\mathrm i y)\)   \(u\!+\!\mathrm i v=\mathrm{SuNem}_1(x\!+\!\mathrm i y)\)   \(u\!+\!\mathrm i v=\mathrm{SuNem}_2(x\!+\!\mathrm i y)\)

For the Nemtsov function \(\mathrm{nem}_q\), the superfunction \(\mathrm{SuNem}_q~\) is real-holomorphic solution \(F\) of the transfer equation \[F(z\!+\!1)=\mathrm{nem}_q\big( F(z)\big)\] with specific asymptotic behaviour at \(-\infty\), namely, \[ F(z)=\frac{1}{\sqrt{-2 z}}\left( 1-\frac{q}{\sqrt{-2 z}} + O\big( \ln(-z)/z\big) \right)\] In order to specify function SuNem, the additional condition is assumed: \[\mathrm{SuNem}_q(0)=1\] (Similar condition is used to specify tetration as superfunction of exponent).

Explicit plot of function SuNem of the real argument is shown in figure at left, \(y\!=\!\mathrm{SuNem}_q(x)\) is plotted versus \(x\) for various values of \(q\).

Function SuNem grows monotonously from zero at \(-\infty\), takes value unity at zero and then grows quickly to infinity for positive values of the argument. The larger is parameter \(q\), the faster is the growth at \(+\infty\). This behaviour corresponds to the intuitive expectations about this function.

Complex maps of function SuNem are shown in figures at right for \(q\!=\!0\), \(q\!=\!1\) and for \(q\!=\!2\); \[u\!+\!\mathrm i v=\mathrm{SuNem}_q(x\!+\!\mathrm i y)\] Maps for different values of \(q\) look similar; however, the greater is \(q\), the faster is the growth of \(\mathrm{SuNem}_q\) along the real axis. This is seen also at the explicit plot in figure at left.

In order to construct function SuNem, first, any superfunction \(F\) with appropriate asymptotic behaviour is constructed.
Then SuNem is set as \[\mathrm{SuNem}_q(z)=F(x_1+z)\] where \(x_1\) is real solution of equation \(F(x_1)=1\).

Abelfunction

\(y\!=\!\mathrm{AuNem}_q(x)\) for \(q=0\), \(1\), \(2\).

For the Abel function of the Nemtsov function, notation AuNem is suggested.

Explicit plot \(y\!=\!\mathrm{AuNem}_q(x)\) versus \(x\) is shown in figure at left for \(q\!=\!0\), \(q\!=\!1\) and \(q\!=\!2\).

The same plot can be obtained, reflecting the curves for SuNem from the angle bisector of the First quadrant of the coordinate plane.

This property can be used as a numerical test of relation \[\mathrm{SuNem}_q(\mathrm{АuNem}_q(x))=x\] it should be so, while \[\mathrm{AuNem}_q=\mathrm{SuNem}_q^{-1}\] As inverse of the superfunction, function AuNem satisfies the Abel equation \[\mathrm{AuNem}_q\big( \mathrm{nem}_q(z)\big)=\mathrm{AuNem}_q(z)+1\] Function AuNem satisfies also the additional condition \[\mathrm{AuNem}_q(1)=0\] that is determined by the corresponding property of the SuNem, namely, that \( \mathrm{SuNem}_q(0)\!=\!1~\).

The asymptotic expansion of function AuNem at zero can be found, inverting the asymptotic expansion of function SuNem at \(-\infty\). The coefficients of this expansion can be found also from the Abel equation.

Iterates

\(y\!=\!\mathrm{nem}_0^{\,n}(x)\) versus \(x\) at various \(n\) \(y\!=\!\mathrm{nem}_1^{\,n}(x)\) versus \(x\) at various \(n\) \(y\!=\!\mathrm{nem}_2^{\,n}(x)\) versus \(x\) at various \(n\)

Using the functions SuNem and AuNem, the \(n\)th iterate of the Nemtsov function can be expressed as follows: \[\mathrm{nem}_q^{\,n}(z)=\mathrm{AuNem}_q\big( n+\mathrm{AuNem}_q(z)\big)\] for
\(q\!=\!0\),
\(q\!=\!1\) and for
\(q\!=\!2\),
these iterates are shown in figures at right.
These iterates look similar to other iterates of the fast growing functions.

At positive number \(n\) of iterate, the iterate shows the fast growth; at negative values of \(n\), the growth is slow.

The zeroth iterate (\(n\!=\!0\)) appears as the identity function.
In the figures, the corresponding graphics are marked with green lines.

Due to the singularity of function ArqNem at zero, the non-integer iterates are not defined at zero and the negative part of the real axis, although they approach zero as the positive argument of the iterate becomes small.

The curves of the iterates show the symmetry with respect to reflections from the bisector of the First quadrant of the coordinate plane, as

\(\mathrm{nem}_q^{-n}(z)=\mathrm{ArqNem}_q^{\,n}(z)\)

Applications

The Nemtsov function had been considered as a candidate of the transfer function, for which the superfunction and the Abel function are difficult to construct with the exotic iterate at its fixed point zero.

Indeed, the construction required certain efforts; they were related mainly with construction of the inverse function ArqNem, the efficient implementation and, especially, with guessing, that namely ArqNem should be declared as the default \(\mathrm{nem}_q^{-1}\) in construction of the Abel function and non-integer iterates, that are holomorphic in the most of the complex plane.

For the Nemtsov function, two other inverse functions have been constructed, they are denoted with names ArcNem and ArkNem.
They have different positions of the cut lines, and, at the iterates, do not provide the real-holomorphic Abelfunction in the wide range as the ArqNem. In such a way, names ArcNem and ArkNem are reserved for the native inverse functions; but the ArqNem is postulated as the default inverse function.

Once the iterates of the inverse function leads to the fixed point of the transfer function, the exotic iterates become straightforward.
Other exotic transfer functions can be considered in the similar way, while their expansion at the fixed point begins with the identity function and the cubic term.
Without loss of generality, the coefficient at the cubic term can be treated as unity; the corresponding transform to this case is shown in the last row of the Table of superfunctions.

Warning

The name «Nemtsov function» is chosen as a simplest mnemonic referring to the event of the 21st century.

The naming is not an attempt to remind the Russian usurper and his crime accomplices that it is not a good style, to kill the political opponent.
The growth of corruption at putin's pahanat indicates that no moral criteria or concepts can be applied to the Russian usurper nor to his accomplices. So, the attempt mentioned would be just vain (even through some impostors clam they can stop war in 24 hours dealing with the war crime and appeasing aggression).

However, Editor keeps the right to call things in the most convenient and efficient system of notations.

Acknowledgement

ChatGPT helped to improve this article.

References

Keywords

«Abel function», «ArqNem», «AuNem», «Book», «C++», «Exotic iterate», «Exotic iteration», «Latex», «Mathematica», «Nemtsov function», «Putin killed Nemtsov», «SuNem», «Superfunction», «Superfunctions», «Table of superfunctions»,

«Немцов Борис Ефимович», «Путин убил Немцова»,