# SuNem

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$$y\!=\!\mathrm{SuNem}_{q}(x)$$ for various $$q$$
$$u\!+\!\mathrm i v=\mathrm{SuNem}_{0}(x\!+\!\mathrm i y)$$
$$u\!+\!\mathrm i v=\mathrm{SuNem}_{1}(x\!+\!\mathrm i y)$$
$$u\!+\!\mathrm i v=\mathrm{SuNem}_{2}(x\!+\!\mathrm i y)$$

SuNem is superfunction for the Nemtsov function

$$\mathrm{Nem}_{q}(z)=z+z^3+qz^4$$

It is assumed, that $$q\!>\!0$$, although the formula can be used for some other values of the parameters too.

SuNem is specific solution of the transfer equation

$$\mathrm{Nem}_{q}\big( \mathrm{SuNem}_{q}(z)\big)=\mathrm{SuNem}_{q}(z\!+\!1)$$.

It is assumed that $$\mathrm{SuNem}_{q}(0)=1$$.

Also, the specific asymptotic behaviour at infinity is assumed,

$$\mathrm{SuNem}_{q}(z) = {\displaystyle \frac{1}{\sqrt{-2z}}}\left(1+O\left(\frac{1}{\sqrt{-2z}} \right)\right)$$

for any fixed phase $$\mathrm{Arg}(z)$$ different from zero.

For any $$\varepsilon>0$$, the formula is valid for any large $$|z|$$ such that $$|\mathrm{Arg}(z)|>\varepsilon$$.

Along the real axis, SuNem shows fast growth from zero at $$-\infty$$ to plus infinity at $$+\infty$$.

## Asymptotic expansion

Function SuNem is constructed by its asymptotical expansion.

For the superfunction $$F$$ of the Nemtsov transfer function $$T=\mathrm{Nem}_{q}$$, it can be obtained from the transfer equation

$$T(F(z))=F(z+1)$$

Keping some positive integer mumber $$M$$ of terms, the asymptotic solution can be written as follows:

$$\displaystyle \tilde F(z) = \frac{1}{\sqrt{-2z}} \left(1+ \frac{P_m(\ln(-z))} {(-2z)^{m/2}} \right)$$

where $$\displaystyle P_m(z)=\sum_{n=0}^{\mathrm{IntegerPart}(m/2)} A[m,n]\, z^m$$

Substitution of $$\tilde F$$ into the transfer equation gives the coefficients $$A$$. These coefficients can be calculated with the mathematica code below:

## Mathematica generator of the algorithm

The first 18 terms of the asymptotic representation of super function $$F$$ can be computed with Mathematica software, using the code below:

T[z_]=z+z^3+q z^4
P[m_, L_] := Sum[a[m, n] L^n, {n, 0, IntegerPart[m/2]}]
a[1, 0] = -q; a[2, 0] = 0;
m = 2;
F[m_,z_] = (-2 z)^(-1/2) (1 + Sum[P[n, Log[-z]]/(-2 z)^(n/2), {n, 1, 2}]);
s[m]=Numerator[Normal[Series[(T[F[m,-1/x^2]] - F[m,-1/x^2+1]) 2^((m+1)/2)/x^(m+3), {x,0,0}]]]
sub[m] = Extract[Solve[s[m]==0, a[m,1]], 1];
SUB[m] = sub[m]

For[m = 3, m < 18,
F[m, z_] = ReplaceAll[(-2 z)^(-1/2) (1+Sum[P[n, Log[-z]]/(-2 z)^(n/2), {n,1,m}]), SUB[m-1]];
s[m] = Numerator[Normal[Series[(T[F[m,-1/x^2]]-F[m,-1/x^2+1]) 2^((m+1)/2)/x^(m+3),{x,0,0}]]];
t[m] = Collect[Numerator[ReplaceAll[s[m], Log[x] -> L]], L];
u[m] = Table[
Coefficient[t[m] L, L^n] == 0, {n, 1, 1 + IntegerPart[m/2]}];
tab[m] = Table[a[m, n], {n, 0, IntegerPart[m/2]}];
Print[sub[m] = Simplify[Extract[Solve[u[m], tab[m]], 1]]];
SUB[m] = Join[SUB[m - 1], sub[m]];
m++];

For[m=1, m<18,
For[n=0,n<(m+1)/2,
A[m, n] = TeXForm[ReplaceAll[a[m, n], sub[m]]];
Print["APQ[", m, "][", n, "]=", A[m, n], ";"]
n++];
m++];

## Evaluation of superfunction

$$y\!=\!\mathrm{SuNe}_q(x)$$ for $$q=-1, -0.5, 0, 0.5, 1, 2, 3$$
displacement $$x_0$$ versus parameter $$q$$

First, the superfunction of the Nemtsov function is constructed, that does not satisfy the requirement on its value at zero. The idea is to use the asymptotival expansion $$\tilde F$$ of the superfunction in the area, where it provides the good approximation, displacing the argument of superfunction into this area with using of the transfer equation.

Superfunction $$\mathrm{SuNe}_q$$ of the Nemtsov function $$\mathrm{Nem}_q$$ appears as limit

$$\displaystyle \mathrm{SuNe}_q(z)=\lim_{n \rightarrow \infty} \mathrm{Nem}_q^{\,n} (\tilde F(z\!-\!n))$$

Explicit plot of function $$\mathrm{SuNe}_q$$ is shown in figure at right for $$q=-1, -0.5, 0, 0.5, 1, 2, 3~$$. Then, function SuNem appears with the appropriate displacement of the argument:

$$\mathrm{SuNem}_q(z)=\mathrm{SuNe}_q(x_0(q)+z)$$

where displacement $$x_0=x_0(q)$$ is real solution of equation $$F_q(x_0)\!=\!1$$. This solution is shown in figure at left. In order to show the general trend of function $$x_0$$, the graphic is extended a little bit into the range of negative $$q$$.

## Inverse function

Inverse function of SuNem is function AuNem, that is Abel function of the Nemtsov function; in wide range of values of $$z$$, it satisfies the Abel equation

$$\mathrm{AuNem_{q}} \big( \mathrm{Nem_{q}}(z)\big) = \mathrm{Nem_{q}}(z\!+\!1)$$

## Iterates of the Nemtsov function

With functions SuNem and AuNem, the iterates of the Nemtsov function can be written as usually:

$$\mathrm{Nem}^n(z)=\mathrm{SuNem}(n+\mathrm{AuNem}(z))$$