File:SqrtExpZ.jpg

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