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<p><b>New page</b></p><div>[[File:SuTraPlo3T.jpg|300px|thumb|Fig.1. $~y\!=\! \mathrm{SuTra}(x)~$ and $~y\!=\!-\! \ln(-x)$]]<br />
[[File:SuTraMapT.jpg|300px|thumb|Fig.2. $u\!+\!\mathrm i v\!=\! \mathrm{SuTra}(x\!+\!\mathrm i y)$]]<br />
<br />
[[SuTra]], or SuperTrappmann function is [[superfunction]] of the [[Trappmann function]], $~\mathrm{tra}(z)=z+\exp(z)~$<br />
<br />
The Trappmann function is treated as the [[transfer function]]; and the [[SuTra]] function, as superfunction of [[tra]],<br />
satisfies the [[transfer equation]]<br />
<br />
(1) $~ ~ ~ \mathrm{SuTra}(z\!+\!1) = \mathrm{tra}\Big( \mathrm{SuTra}(z)\Big)$<br />
<br />
In orfer to narrow the set of solutions, it is assumed, that <br />
<br />
(2) $~ ~ ~ \mathrm{SuTra}(z^*) = \mathrm{SuTra}(z)^* ~ ~$, $~ ~ ~ \mathrm{SuTra}(0) = 0 $<br />
<br />
and that SuTra is [[entire function]]. Also, the logarithmic asymptotic bahavior at infinity is assumed, at least outside the strip <br />
<br />
(3) $~ ~ ~ \{ z\in \mathbb Z : \Re(z)\!>\!2, |\Im(z)|\!<\!2 \}$<br />
<br />
Function [[SuTra]] is expected to be the only function with such a behavior.<br />
The plot of the [[SuTra]] is shown in Figure 1. For comparison, its asymptotic $~u\!=\!-\!\ln(-x)~$ is shown with thin line.<br />
The [[complex map]] of the [[SuTra]] is shown in Figure 2.<br />
<br />
==History==<br />
<br />
The [[Trappmann function]], $~\mathrm{tra}(z)=z+\exp(z)~$, had been suggested in 2011 by [[Henryk Trappmann]] as an example of a [[transfer function]], for which it is difficult to construct the [[superfunction]]. For this reason, the [[Trappmann function]] is called after his name.<br />
<br />
However, the [[superfunction]] of the [[Trappmann function]], id est, [[SuTra]] can be expressed through that of the function [[zex]], $~\mathrm{zex}(z)=\mathrm{ArcLambertW}(z)=z\,\exp(z)$, in the following way:<br />
<br />
(4) $~ ~ ~ \mathrm{SuTra}(z)=\ln\Big( \mathrm{SuZex}(z)\Big)$<br />
<br />
at least for $z$ outside the strip (3). Perhaps, the similar representation could be used also for values inside that strip with appropriate choice of branch of the logarithmic function. Practically, it is easier to use the [[transfer equation]] (1) recurrently, bringing the argument of SuTra to the range there the representation (4) is valid. In such a way function SuTra is implemented in the [[C++]] algorithm [[SuTra.cin]] for plotting of the [[complex map]] in Figure 1.<br />
<br />
==Iteration of the [[Trappmann function]]==<br />
The Trappmann function is important example of function without [[fixed points]]. Such functions were expected to be difficult for construction of the [[fractional iterate]]s. However, due to expression (4), the iterates of the [[Trappmann function]] can be expressed with general formula<br />
<br />
(5) $~ ~ ~ \mathrm{Tra}^n(z)=\mathrm{SuTra}\Big( n+ \mathrm{AuTra}(z)\Big)$<br />
<br />
where [[AuTra]]$\,=\!\mathrm{SuTra}^{-1}$ is the [[Abel function]] for the [[Trappmann function]]; the number $n$ of iteration in expression (5) has no need to be integer. As usually, the [[superfunction]] can be iterated any real (can even complex) number of times.<br />
<br />
In certain range of values of the argument, and, in particular, along the real axis,<br />
The Abelâ€“Trappmann function [[AuTra]] can be expressed through the [[Abel function]] of [[zex]], id est, through [[AuZex]] in the following way:<br />
<br />
(6) $~ ~ ~ \mathrm{AuTra}(z)=\mathrm{AuZex}\Big(\exp(z)\Big)$<br />
<br />
Representations (6) and (5) are used to plot graphics of the [[nonâ€“integer iterate]]s of the [[Trappmann function]].<br />
<br />
==References==<br />
<references/><br />
<br />
==Keywords==<br />
[[Trappmann function]],<br />
[[SuperFunction]],<br />
[[Zex]],<br />
[[SuZex]],<br />
[[AuTra]],<br />
[[Iteration]]<br />
<br />
[[Category:SuTra]]<br />
[[Category:Trappmann function]]<br />
[[Category:Superfunction]]<br />
[[Category:Articles in English]]</div>Maintenance script