Difference between revisions of "File:B271a.png"

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[[Complex map]] of [[ArcTetration]] to base $\mathrm e \approx 2.71~$:
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function $f=\mathrm{ate}(x+\mathrm i y)$ is shown in the $x$,$y$ plane with levels $u\!=\!\Re(f)\!=\!\mathrm{const}$ and
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levels $v\!=\!\Im(f)\!=\!\mathrm{const}$.
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The branchpoints $L\!\approx \! 0.2+1.3 \mathrm i$ and $L^*$ are [[fixed point]]s of [[logarithm]].
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The cut lines go along the half-lines $y\!=\! \pm \Im(L)$, $x\!<\! \Re(L)$.
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This choice of cut lines differs from that used in the first description of such function <ref name="analuxp">
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http://tori.ils.uec.ac.jp/PAPERS/2009analuxpRepri.pdf
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D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670. <!--
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http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html !-->
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</ref>.
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The ArcTetration $\mathrm{ate}$ is inverse of [[tetration]] and holomorphic solution of the [[Abel equation]]
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:$ \mathrm{ate}(\exp(z))=\mathrm{ate}(z)\!+\!1~,$
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:$ \mathrm{ate}(1)=0.$
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Such a soluton is shown to be unique
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<ref name="uniabel">
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http://tori.ils.uec.ac.jp/PAPERS/2011uniabel.pdf
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H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)<!--
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http://www.springerlink.com/content/u7327836m2850246/ !-->
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</ref>.
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The fast [[C++]] implementation of such a function is available below. The construction of such implementation is desctibed in <ref name="vladi">
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http://tori.ils.uec.ac.jp/PAPERS/2010vladie.pdf
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D.Kouznetsov. Superexponential as special function. [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-45.
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</ref><ref name="cztetration">
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http://en.citizendium.org/wiki/Tetration
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</ref>.
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==Generation of the image==
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The following files are used for generation of the [[complex map]].
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[[Fslog.cin]] , the routine in [[C++]], $\rm complex <double> FSLOG(complex <double> z) $
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is used. For generation, it should be saved as "fslog.cin"
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[[Conto.cin]] , the routine to make the [[contour plot]]. It should be saved as "conto.cin"
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[[B271az.cc]] , the [[C++]] code that includes the *.cin mentioned above and draws the mesh of the [[complex map]], writing the output file "b271az.eps" and "b271az.pdf"
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[[B271a.tex]] , the [[Latex]] code that adds the labels and, at the compilation, gives "b271a.pdf". The last was converged to the [[PNG]] format with resolution 150pixels per inch.
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The [[C++]] compiler and the [[Latex]] compilers should be installed at the computer. The image was generated at [[Macintosh]], but very similar images were generated also in [[Debian]]. Hope, the same generators can be used with other operational systems too.
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The files mentioned above are in preparation for the uploading and should appear soon.
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For future: should all the uploaded generators have names beginning from the capital letter?
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==References==
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<references/>
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[[Category:Complex maps]]
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[[Category:Tetration]]

Revision as of 09:39, 21 June 2013

Complex map of ArcTetration to base $\mathrm e \approx 2.71~$: function $f=\mathrm{ate}(x+\mathrm i y)$ is shown in the $x$,$y$ plane with levels $u\!=\!\Re(f)\!=\!\mathrm{const}$ and levels $v\!=\!\Im(f)\!=\!\mathrm{const}$.

The branchpoints $L\!\approx \! 0.2+1.3 \mathrm i$ and $L^*$ are fixed points of logarithm.

The cut lines go along the half-lines $y\!=\! \pm \Im(L)$, $x\!<\! \Re(L)$.

This choice of cut lines differs from that used in the first description of such function [1].

The ArcTetration $\mathrm{ate}$ is inverse of tetration and holomorphic solution of the Abel equation

$ \mathrm{ate}(\exp(z))=\mathrm{ate}(z)\!+\!1~,$
$ \mathrm{ate}(1)=0.$

Such a soluton is shown to be unique [2].

The fast C++ implementation of such a function is available below. The construction of such implementation is desctibed in [3][4].

Generation of the image

The following files are used for generation of the complex map.

Fslog.cin , the routine in C++, $\rm complex <double> FSLOG(complex <double> z) $ is used. For generation, it should be saved as "fslog.cin"

Conto.cin , the routine to make the contour plot. It should be saved as "conto.cin"

B271az.cc , the C++ code that includes the *.cin mentioned above and draws the mesh of the complex map, writing the output file "b271az.eps" and "b271az.pdf"

B271a.tex , the Latex code that adds the labels and, at the compilation, gives "b271a.pdf". The last was converged to the PNG format with resolution 150pixels per inch.

The C++ compiler and the Latex compilers should be installed at the computer. The image was generated at Macintosh, but very similar images were generated also in Debian. Hope, the same generators can be used with other operational systems too.

The files mentioned above are in preparation for the uploading and should appear soon.

For future: should all the uploaded generators have names beginning from the capital letter?

References

  1. ↑ http://tori.ils.uec.ac.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  2. ↑ http://tori.ils.uec.ac.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)
  3. ↑ http://tori.ils.uec.ac.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  4. ↑ http://en.citizendium.org/wiki/Tetration

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