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[[Nemtsov function and its iterates]] is article about the [[Nemtsov function]], adaptation from version, prepared for publication, with goal to check the cross-references and to catch misprints, if any.

<b>Author:</b> Dmitrii Kouznetsov.

<b>Abstract:</b>
The Nemtsov function appears as  polynomial  
\(~\mathrm{Nem}_q(z)=z+z^3+qz^4 \,\); \(~\) 
\(q\)  is parameter. 
The Superfunction, Abelfunction and iterates \(\mathrm{Nem}_q^n\)  for complex \(n\) are constructed.

<b>Keywords:</b> [[Nemtsov function]], [[Superfunction]], [[Abelfunction]], [[Iterate]]

==Introduction==

===Overview===

In 1950, the interest to the non-integer iterates had been boiled-up with iterates of exponential and, in particular, iterate half of the exponential <ref name="knezer">
http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002175851&IDDOC=260718
<!-- \url{http://mizugadro.mydns.jp/PAPERS/Relle.pdf !-->
Helmuth Kneser. Reelle analytische Lösungen der Gleichung 
\(\varphi(\varphi(x))\!=\!e^x\) und verwandter Funktionalgleichungen. Journal für die reine 
und angewandte Mathematik 187 (1950) 56-67
</ref>, id est, function \(\varphi\) such that 
\(\varphi(\varphi(z))\!=\!\exp(z)\). The problem of iteration 
<ref name="seekers">
http://link.springer.com/article/10.1007%2FBF02559539
G. Szekeres. Regular iteration of real and complex functions. Acta Mathematica, September 1958, Volume 100, Issue 3, pp 203-258.
</ref><ref name="walker">
http://www.sciencedirect.com/science/article/pii/0022247X9190029Y
Peter L Walker. On the solutions of an Abelian functional equation.
Journal of Mathematical Analysis and Applications,
Volume 155, Issue 1, February 1991, Pages 93-110.
</ref><ref name="berg">
http://www.ams.org/journals/bull/1993-29-02/ S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf
W.Bergweiler. Iteration of meromorphic functions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188.</ref><ref name="domsta">
http://link.springer.com/article/10.1007%2FBF01831152
Joachim Domsta. A limit formula for regular iteration groups. Aequationes mathematicae, February 1996, Volume 51, Issue 1, pp 207-208.
</ref>
of holomorphic function had been formulated, although 
until year 2009, no efficient algorithm for 
computation of non-integer iterates (except few special functions) had been suggested. 
Then, such algorithms had been reported; and not only for the exponent to 
various 
bases 
<ref name="analuxp">
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxp.pdf 
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
</ref><ref name="vladie">
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. 
</ref><ref name="sqrt2">
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
</ref><ref name="e1e">
http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf
http://www.ils.uec.ac.jp/~dima//PAPERS/2012e1eMcom2590.pdf
http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf
H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation. v.81 (2012), p. 2207-2227. %ISSN 1088-6842(e) ISSN 0025-5718(p) 
</ref><ref name="pen">
http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf <br>
\url{http://www.ils.uec.ac.jp/~dima/PAPERS/2014acker.pdf
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314. </ref><ref name="kkms">
http://kkms.org/index.php/kjm/article/view/428
William Paulsen. Finding the natural solution to f(f(x))=exp(x). Korean J. Math. 24 (2016), No.1, pp.81–108.
</ref><ref name="springer">
http://link.springer.com/article/10.1007/s10444-017-9524-1
William Paulsen and Samuel Cowgill. Solving \(F(z\!+\!1)=b^F(z)\) in the complex plane. Advances in Computational Mathematics, 2017 March 7, p. 1–22
</ref><ref name="cow">
https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1?pq-origsite=gscholar&cbl=18750&diss=y
Cowgill, Samuel. Exploring Tetration in the Complex Plane. Arkansas State University, ProQuest Dissertations Publishing, 2017. 10263680.
</ref><ref name="PaulsenComplex">
https://doi.org/10.1007/s10444-018-9615-7
Paulsen, William. Tetration for complex bases. Advances in Computational Mathematics (2018): 1-25.
</ref>,
but also for other holomorphic 
functions: for factorial by <ref name="fac">
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 <br>
http://mizugadro.mydns.jp/PAPERS/2010superfae.pdf} (English version)<br>
http://mizugadro.mydns.jp/PAPERS/2010superfar.pdf} (Russian version)<br>
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. 
Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14)
</ref>,
for the ``logistic operator'' \(z \!\mapsto\! s\,z\,(1\!-\!z)\) by 
<ref name="logi>
http://www.springerlink.com/content/u712vtp4122544x4} (Official version, DOI 10.3103/S0027134910020049)<br>
http://mizugadro.mydns.jp/PAPERS/2010logistie.pdf (English version)<br>
\url{http://mizugadro.mydns.jp/PAPERS/2010logistir.pdf Russian version)<br>
D.Kouznetsov. Holomorphic extension of the logistic sequence. 
Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31) 
</ref>
for the [[Trappmann function]] \(z \!\mapsto\! z\!+\!\exp(z)\) and function \(z \!\mapsto\! z \exp(z)\) by 
<ref name="hikari">
http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf<br>
http://mizugadro.mydns.jp/PAPERS/2013hikari.pdf<br>
D.Kouznetsov. Entire function with logarithmic asymptotic. 
Applied Mathematical Sciences, 2013, v.7, No.131, p.6527-6541. 
</ref>
and for sin by <ref name="sin">
http://www.pphmj.com/references/8246.htm<br>
http://mizugadro.mydns.jp/PAPERS/2014susin.pdf<br>
http://www/ils.uec.ac.jp/~dima/PAPERS/2014susin.pdf<br>
<!--http://mizugadro.mydns.jp/PAPERS/2014susinL.pdf !-->
D.Kouznetsov. Super sin. 
Far East Journal of Mathematical Science, v.85, No.2, 2014, pages 219-238.
</ref>.

For various examples of a holomorphic functions \(T\), called [[transfer function]], the [[iterate]]s
can be expressed (and evaluated) through the [[superfunction]] \(F\), which is solution of the transfer equation 

\(\!\!(1)~~
T(F(z))=F(z\!+\!1)
\)

<!-- \eL{TF} !-->
and the corresponding [[abelfunction]] \(G=F^{-1}\):

\(\!\!(2)~~
T^n=F\big(n\!+\!G(z)\big)
\) <!--\eL{Tn} !-->

Here, the superscript after the name of function indicates the number of its iterate;
this notation had been suggested in 1993 by W.Bergweiler 
<ref name="berg">
http://www.ams.org/journals/bull/1993-29-02/ S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf
W.Bergweiler. Iteration of meromorphic functions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188.</ref>. In these notations, <!--
%in wide range of \(z\), 
%, at least for real-holomorphic growing function \(T\), !-->

\(\!\!(3)~~T^0(z)=z\)<br>
\(\!\!(4)~~T^1(z)=T(z)\)<br>
\(\!\!(5)~~T^2(z)=T(T(z))\)<br>
\(\!\!(6)~~T^3(z)=T(T(T(z)))\)

and so on; <!--%For example, in these notations, !-->
in particular, \(\sin^2(x)\) denotes \(\sin(\sin(x))\), but neither \(\sin(x)^2\) nor \(\sin(x^2)\). 
In the representation \rf{Tn}, the number \(n\) of iterate has no need to be integer. It can be a fractal and even a complex number. <!--%The holomorphic function can be iterated complex number of times. !-->

The abelfunction \(G=F^{-1}\) satisfies the Abel equation

\(\!\!(7)~~
G(T(z))=G(z)+1\)
<!--\eL{GT} !--
and can be considered as "primary". For the evaluation of \(F\) and \(G\), both equations, 
(1) and (7) are useful <!--
\rf{TF} and \rf{GT} are useful.
\JP{
Superfunction \(F\) appears as solution of the transfer equation
\be
T(F(z))= F(z\!+\!1)
\eL{TF}
and the Abel function \(G=F^{-1}\) satisfies the Abel equation
\be
G(T(z))=G(z)+1
\eL{GT}
}%JP !-->

In addition to equations (1) and/or (7),
<!-- \rf{TF} and/or \rf{GT}, !-->
some supplementary requirements on the asymptotic behaviour of \(F\) and/or \(G\) are applied in order to provide the uniqueness <!-- \cite{uniabel,kkms,springer}. !-->
<ref name="kkms"/><ref name="springer"/><ref name="uniavel"> 
http://www.springerlink.com/content/u7327836m2850246/
http://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf
H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)
</ref>.

Alternatively, the explicit way of the computation may be postulated in the definition of the superfunction \(F\) and/or the Abel function \(G\);
then, this way of computation determines the asymptotic behaviour.

The success of construction of superfunctions for various transfer functions provoke the attempts to construct a holomorphic transfer function \(T\) such that its iterates cannot be determined in natural intuitive way, nor the efficient algorithm for the superfunction and abelfuctions can be constructed.
Consideration of the [[Nemtsov function]] describes the failure of such an attempts; the superfunciton and the abelfunction for the [[Nemtsov function]] can be defined, calculated and supplied with the efficient numerical implementation for the evaluation.

===Previously published methods: [[Regular iteration]]===

For the case of a real-holomorphic transfer function \(T\) with real [[fixed point]] \(L\), (id est, \(T(L)\!=\!L\)), such that
\(T'(L)>0\), the most important construction is the [[Regular iteration]].

Keeping in mind the reading by the colleagues, who did not read the previous articles (an in order not to force the reader to dig the previous publications), the regular iteration is shortly repeated here.

Search for the asymptotic solution of the [[transfer equation]] (1) in the following form:

\( \displaystyle F(z) = L+\sum_{n=1}^{N} a_n \varepsilon^n + o(\varepsilon^N)\) 

where \(N\) is natural number, \(\varepsilon=\exp(kz)\) 
for some constant \(~k~\) and \(~a~\) are constant coefficients.

The substitution into the [[transfer equation]] and the asymptotic analysis 
at \(\varepsilon \ll 1\) gives

\(k=\log(K)=\log(T′(L))\)

and set of equations for coefficients \(a\). 
It is convenient to set \(a_1=1\)
Then, other coefficients appear as solutions of equations

\(a_2 K - a_2 K^2 + {T\,''(L)}/{2} =0 \)

\(a_3 K - a_3 K^2 + 2 a_2  {T\,''(L)}/{2}+ {T\,'''(L)}/{6} = 0\)
<!--
\(a_4 K - a4 K^4 + a2^2 p2 + 2 a3 p2 + 3 a2 p3 + p4\)
!-->

and so on. 

The solution is singular at \(K=\log(T′(L)=0\).

Namely this case is of interest in this article. The Nemtsov function is example of such function:

\(\mathrm{Nem}_q(z)=z+z^3+qz^4\)

\(L=0\),

\(\mathrm{Nem}_q'(L)=1\)

===Exotic iteration: \(T'(L)=1\), \(T''(L) \ne 0\)===

At the [[regular iteration]] of a real–holomorphi function \(T\) the expansion of the superfunction \(F\) begins with constant (fixed point \(L\)) and the exponential term.

I the case \(T'(L)=0\) the expansion of super function begins with with constant (fixed point \(L\)) and [[power function]], and also includes the logarithmic term. This can be written as follows:

\(T(L)=L\)

\(\displaystyle
F[z] \sim L + \frac{a}{z}+ \sum_{m=1}^M \frac{P_m(\ln(\pm z))}{z^{m+1}}+O\left(\frac{\ln(\pm z)^{M+1}}{z^{M+1}}\right)\)

for any positive integer \(M\). Here, \(P_m\) is some polynomial of \(m\)-th order. Coefficients of this polynomials can be calculated, substituting the asymptotic representation above into the transfer equation. Sign + or - should be chosen, dependently, should the iterate \(T^n(z)\) be holomorphic 
at \(z>L\) to at \(z<L\).
In general cases, both seem to be impossible; at the fixed point \(L\), the non–integer iterate have the branch point;
In order to get iterates \(T^n(z)\), growing at \(z>L\), we should choose sign -. Then, truncation of the asymptotic expansion at some \(M\) gives the approximation, valid for large negative values of the real part of the argument. From these values, the superfunction can be evaluated with any required precision. Then, the solution can be extender to the whole complex plane, iterating the transfer equation. 
In such a way, the asymptotic solution determines (and gives way for the efficient evaluation) of the superfunction.

For case \(T=\exp^2(-1)\), id est \(T(z)=\exp(z/e)\), the superfunctions, abelfunctions and iterate are described in 2011
<ref name="e1e"/>.

The leasing coefficient \(a\) in the expansion above is expressed as follows:

\(a=-2/T''(L)\)

In such a way,  for this method, condition \(T''(L) \ne 0\) is essential.

===Exotic iteration: \(T'(L)=1\), \(T''(L) = 0\), \(T'''(L) \ne 0\)===
The restriction on the second derivative at the fixed point indicates the candidate for the transited function, that is supposed to be difficult to iterate. The simples case is \(T'(L)=1\), \(T''(L) = 0\), \(T'''(L) \ne 0\).

Actually, this case is not very exotic; this takes place, for example, for \(T=\sin\), considered in 2014; the superfunction [[SuSin]] and abelfunction [[AuSin]] are constructed <ref name="sin"/>:

\(T(L)\!=\!L\!=\!0\), \(T'(L)\!=\!1\), \(T''(L)\! =\! 0\), \(T'''(L) \!=\!-1/6\)

The consideration of sin, and the straightforward generalisation for other functions \(T\) with non–zero \(T'''(L)\) <ref name="sin"/> explicitly uses the symmetry \(T(-z)=-T(z)\).

The "minimal" distortion, of the symmetry seemed to give an example to break the pretentious statement<br>
<b>for any real-holomorphic transfer function, the superfunction, abelfunction and non–integer iterates can be constructed!</b>

The Nemtsov function seemed to be the simplest example of function of such a kind.
For this reason, the Nemsov function 

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

is especially interesting.

This gives sway to iterate also some other polynomials, that are expressed through the Nemtsov function with linear conjugation

\(T(z)= U(\mathrm{Nem}_q(V(z)))\)

while \(V=U^{-1}\); the simple conjugation appear with

\(U(z)=\mu+\nu z\)

\(V(z)=(z-\mu)/\nu\)

For this reason, the coefficients at the linear and cubic term in the Nemtsov function are chosen to be unity.
As for the 4th order coefficient, it is not so easy to adjust with the linear conjugation; so, is is kept as a free parameter.
In particular, it helps to reproduce results for the symmetric function (that had been implemented previously) and serves as a test of the algorithm of calculation of the Nemtsov function and its iterates.

==Inverse function==
Inversion of the Nemtsov function is not so easy, as it seems to be.
At given \(x\), The 4th order equation

\(y+y^3+q y^4=x\)

has 4 solutions. In order to avoid unwanted cut lines in the maps of the Abelfunction and iterates of the Nemtsov function, 
the branches for the inverse function should be chosen. In particular, it is convenient, that the cut lines are straight line.
Any of solutions, provided with code

Solve[y + y^3 + q y^4 == x, y]

has curvilinear cut lines.
For this reasons, the 3 inverse functions are constricted:

[[ArcNem]]\(_q\)

[[ArkNem]]\(_q\)

[[ArqNem]]\(_q\)

They coincide in vicinity of the positive part of the real axis, but have different cuts in the complex plane.

==References==
<references/>