File:QFactorialQexp.jpg

Functions sqrt(!) , left, and sqrt(exp), right, in the complex plane. For a transfer function \(H\), the superfunction \(S\) is a holomorphic solution of the functional equation \[ H\big(S(z)\big)=S(z\!+\!1)\] In the cases considered, \(H\!=\!\mathrm{Factorial}\), left, and \(H\!=\!\exp\), right. The corresponding functions \(S\) are, respectively, SuperFactorial, left, and tetration, right. The fractional power (id eat, the fractional iteration) \(H^c\) of the transfer function \(H\) is expressed with \[ H^c(z)=S(c+S^{-1}(z))\] For \(c=1/2\) this expression determines the function \(\sqrt{H}\). Functions \(f=\sqrt{!}(z)\) and \(f=\sqrt{\exp}(z)\) are shown in the complex \(z\) plane with levels \(p=\Re(f)=\mathrm{const}\) and levels \(q=\Im(f)=\mathrm{const}\).

Levels \(p=-4,-3,-2,-1,0,1,2,3,4\) are shown with thick black lines.

The intermediate levels for \(p<0 \) are shown with thin red lines.

The intermediate levels for $p>0 $ are shown with thin blue lines.

Levels $q=-4,-3,-2,-1$ are shown with thick red lines.

Levels $q=-1,2,3,4$ are shown with thick blue lines.

The intermediate levels $q=\mathrm{const}$ are shown with thin green lines.

The cuts of the range of holomorphism are shown with black dashed lines.

This image is copied from http://en.citizendium.org/wiki/Image:QFacQexp.jpg ; it is published at http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 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). Please indicate the source at the reuse.