Nemtsov function



Nemtsov function is polynomial of special kind, suggested as an example of a transfer function for the book Superfunctions.

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

Complex map of \(\mathrm{Nem}_q\) is shown in figure 2 for \(q\!=\!0\) and in figure 3 for \(q\!=\!2\) with lines \(u\!=\!\mathrm{const}\) and lines \(v\!=\!\mathrm{const}\) assuming that \(u\!+\!\mathrm i v=\mathrm{Nem}_0(x\!+\!\mathrm i y)\).

Definition
Let \(q\) be non–negative real parameter. Then, Nemtsov Function \(\mathrm{Nem}\) is defined for complex argument \(z\) as follows:

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

Notations
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 cannot be applied "as is", and some modification, generalisation is required. The need of this modification had been revealed 2015.02.27, in the day, when Putin killed Nemtsov. Since February to July of 2015, no other scientific concept of that murder hat been suggested. Apparently, the total corruption in Russia does not allow the professional to investigate that case, and that terroristic act caused many publications. This makes name Nemtsov to be a good mark, label on the timeline of Hunan History.

For this reason, function \(\mathrm{Nem}_q\) by equation (1) is called after the name of Boris Nemtsov.

In this article, properties of the Nemtsov function are considered, 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,

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

Abel function, denoted with SuNem,

\(\mathrm{SuNem}_q(\mathrm{Nem}_q(z))=\mathrm{SuNem}_q(z)+1\)

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 corresponding Abel function; 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.

General properties
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 monotonously. The monotonous growth has also the inverse function ArqNem, the Abel function AuNem and the real iterates of the Nemtsov function.

As for any non–trivial entire 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; at \(z=\mathrm{NemBran}(q)\), function \(\mathrm{ArqNem}_q(z)\) has infinite derivative.

For construction of the inverse function, important are the complex soluitons \(A\) of equation \(\mathrm{Nem}_q^{\prime}=0\). One of these solutions is expressed with function NemBra, id est, \(A=\mathrm{NemBra}(q)\). Parametric plot of this function is shown in figure 4:

\(x=\mathrm{Re}\big(\mathrm{NemBra}(q)\big)~\)

\(y=\mathrm{Im}\big(\mathrm{NemBra}(q)\big)~\)

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

Inverse function
The Nemtsov function has various inverse functions, as equation

\(\mathrm{Nem}_q(A)=z\)

has many solutions \(A\) ; there are three solutions for \(q\!=\!0\) and four for \(q\!\ne\!0\). The holomorphic inverse function unavoidably has the cut lines. The appropriate choice of these cut lines happens to be non–trivial problem.

For construction of the Abel function, denoted as AuNem, the special inverse function ArqNem of the Nemtsov function is chosen. For function ArqNem, the cut lines are chosen in the following way. One cut line goes along the negative part of the real axis from \(-\infty\) to zero. Two other cut lines go straightly from zero to the complex branchpoints \(Z_o\) of the inverse function. These brach points can be expressed in the following way:

\(Z_{\mathrm o}=\mathrm{Nem}_q(z_{\mathrm o})\)

where \(z_{\mathrm o}=\mathrm{NemBran}(q)\) is solution of equation

\(\mathrm{Nem}_q^{\prime}( z_{\mathrm o})=0\)

The following relation takes place:

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

Parametric plot of function NemBran is shown in figure 5:

\(x=\mathrm{Re}\big(\mathrm{NemBran}(q)\big)~\),

\(y=\mathrm{Im}\big(\mathrm{NemBran}(q)\big)~\)

At small values of the argument, the Nemtsov function looks similar to the identity function; so, the parametric plot of function NemBran in figure 5 looks similar to that of function NemBra in figure 4.

Complex map of function ArqNem is shown in figure 6 for \(q\!=\!0~\) and in figure 7 for \(q\!=\!2~\):

\(u\!+\!\mathrm i v= \mathrm{ArqNem}_q(x\!+\!\mathrm i y)\)

The complex double implementation of function ArqNem is loaded as arqnem.cin ; parameter \(q\) is storem in the global variable \(Q\). 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


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 infinity, namely,

\(\displaystyle F(z)=\frac{1}{\sqrt{-2 z}}\left( 1-\frac{q}{\sqrt{-2 z}} + O\big( \ln(-z)/z\big) \right)\)

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, \(\mathrm{SuNem}_q(z)=F(x_1+z)\) where \(x_1\) is real solution of equation \(F(x_1)=1\).

Abel function
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 bisektris of the First quadrant of the coordinate plane. This property can be used as a mumerical test of relation \(\mathrm{SuNem}_q(\mathrm{АuNem}_q(x))=x\); it should be so, as

\(\mathrm{AuNem}_q=\mathrm{SuNem}_q^{-1}\)

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 coeffcieints of this expansion can be found also from the Abel equation. The expansion can be written as follows:

Iterates of the Nemtsov function


With 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 (\(n\!=\!0\)), the iterates apperas as identity function. In the figures, the corresponding graphic is marked with green line.

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 bisectris of the First quadrant of the coordinate plane, as

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

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

Indeed, the construction required certain efforts, but they sere related mainly with construction of the inverse function ArqNem, the efficient implementation and, especially, with guessing, that namely ArqNem shouls be used to get 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 hames ArcNem and ArkNem. They have different positions of the cut lines, and, at the iterates, do not provide the holomorphic Abel function.

Once the iterates of the inverse function leads to the fixed point of the transfer function, the exotic iterates are straightforward. Other transfer functions can be considered in the similar way, assuming, that 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.

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