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.