Difference between revisions of "File:SqrtExpZ.jpg"

From TORI
Jump to navigation Jump to search
(Importing image file)
 
Line 1: Line 1:
  +
[[Complex map]]s of function <math>f=\exp^c(z)</math> in the <math>z</math> plane
−
Importing image file
  
  +
for some real values of <math>c=\pm 1, \pm 0.9, \pm 0.5, \pm 0.1</math>.
  +
Levels
  +
<math>\Re(f) =-3,-2,-1,0,1,2,3,4,5,6,7,8,9</math> and
  +
<math>\Im(f) =-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14</math> are shown with thick lines.
  +
Red means negative values and blue indivates positive values of the real or imaginary part.
  +
  +
Also, levels <math>\Re(f) = 0.31813150520476413</math> and
  +
<math>\Im(f) =1.3372357014306895</math>
  +
are shown with thick green lines. The intersection of these lines indicate the fixed point $L$, which is solution of $\ln(L)=L$ ; this points maps to itself.
  +
  +
The thin lines indicate intermediate levels.
  +
Pink lines indicate the cuts of the range of [[holomorphism]].
  +
  +
The function is real-holomorphic, so, the maps are symmetric, and tho only upper half of the complex plane is shown for each case.
  +
  +
The primary algorithm of the evaluation is described in the [[Mathematics of Computation]]
  +
<ref name="moc">
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
  +
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  +
</ref>;
  +
  +
$\exp^c(z)=F(c+G(z))$ ,
  +
  +
where $F$ is [[tetration]] and $G=F^{-1}$ is the arctetrational (called also super-logarithm or the the Abel–exponential).
  +
  +
The complex double implementation used to boost the evaluation is described in the [[Vladikavkaz Mathematical Journal]]
  +
<ref name="vladi">
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf
  +
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  +
</ref>.
  +
  +
Copyleft 2008 by Dmitrii Kouznetsov: Use for free, attribute the source.
  +
  +
This image appears also at
  +
http://en.citizendium.org/wiki/File:Sqrt(exp)(z).jpg
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Tetration]]
  +
[[Category:Superfunctions]]
  +
[[Category:Complex maps]]
  +
[[Category:Mathematics]]

Revision as of 11:55, 21 June 2013

Complex maps of function \(f=\exp^c(z)\) in the \(z\) plane for some real values of \(c=\pm 1, \pm 0.9, \pm 0.5, \pm 0.1\). Levels \(\Re(f) =-3,-2,-1,0,1,2,3,4,5,6,7,8,9\) and \(\Im(f) =-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14\) are shown with thick lines. Red means negative values and blue indivates positive values of the real or imaginary part.

Also, levels \(\Re(f) = 0.31813150520476413\) and \(\Im(f) =1.3372357014306895\) are shown with thick green lines. The intersection of these lines indicate the fixed point $L$, which is solution of $\ln(L)=L$ ; this points maps to itself.

The thin lines indicate intermediate levels. Pink lines indicate the cuts of the range of holomorphism.

The function is real-holomorphic, so, the maps are symmetric, and tho only upper half of the complex plane is shown for each case.

The primary algorithm of the evaluation is described in the Mathematics of Computation [1];

$\exp^c(z)=F(c+G(z))$ ,

where $F$ is tetration and $G=F^{-1}$ is the arctetrational (called also super-logarithm or the the Abel–exponential).

The complex double implementation used to boost the evaluation is described in the Vladikavkaz Mathematical Journal [2].

Copyleft 2008 by Dmitrii Kouznetsov: Use for free, attribute the source.

This image appears also at http://en.citizendium.org/wiki/File:Sqrt(exp)(z).jpg

References

  1. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html 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://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current17:50, 20 June 2013Thumbnail for version as of 17:50, 20 June 2013842 × 953 (83 KB)Maintenance script (talk | contribs)Importing image file

You cannot overwrite this file.

The following file is a duplicate of this file (more details):

The following 3 pages use this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:SqrtExpZ.jpg&oldid=5678"