File:QFacQexp.jpg

For a transfer function ${\displaystyle H}$, the superfunction ${\displaystyle S}$ is a holomorphic solution of the functional equation

${\displaystyle H{\big (}S(z){\big )}=S(z\!+\!1)}$

In the cases considered, ${\displaystyle H\!=\!\mathrm {Factorial} }$, left, and ${\displaystyle H\!=\!\exp }$, right. The corresponding functions ${\displaystyle S}$ are, respectively, SuperFactorial, left, and tetrational, right. The fractional power (id eat, the fractional iteration) ${\displaystyle H^{c}}$ of the transfer function ${\displaystyle H}$ is expressed with

${\displaystyle H^{c}(z)=S(c+S^{-1}(z))}$

For ${\displaystyle c=1/2}$ this expression determines the function ${\displaystyle {\sqrt {H}}}$. Functions ${\displaystyle f={\sqrt {!}}(z)}$ and ${\displaystyle f={\sqrt {\exp }}(z)}$ are shown in the complex ${\displaystyle z}$ plane with levels ${\displaystyle p=\Re (f)=\mathrm {const} }$ and levels ${\displaystyle q=\Im (f)=\mathrm {const} }$.

Levels ${\displaystyle p=-4,-3,-2,-1,0,1,2,3,4}$ are shown with thick black lines.

The intermediate levels for ${\displaystyle p<0}$ are shown with thin red lines.

The intermediate levels for ${\displaystyle p>0}$ are shown with thin blue lines.

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

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

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

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

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