File:QFacQexp.jpg: Difference between revisions

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 == Summary ==
-{{Image_Details|user
+Importing file
-|description  = Functions sqrt(!) , left, and sqrt(exp), right, in the complex plane.
-For a [[transfer function]] <math>H</math>, the [[superfunction]] <math>S</math>
-is a [[holomorphic function|holomorphic]] solution of equation
-: <math> H\big(S(z)\big)=S(z\!+\!1)</math>
-In the cases considered, <math>H\!=\!\mathrm{Factorial}</math>, left, and <math>H\!=\!\exp</math>, right.
-The corresponding functions <math>S</math> are, respectively, [[SuperFactorial]], left, and
-[[tetrational]], right.
-The fractional power (id eat, the fractional iteration) <mathh>H^c</math> of the transfer function <math>H</math> is expressed with
-: <math> H^c(z)=S(c+S^{-1}(z))</math>
-For <math>c=1/2</math> this expression determines the function <math>\sqrt{H}</math>.
-Functions <math>f=\sqrt{!}(z)</math> and <math>f=\sqrt{\exp}(z)</math> are shown in the complex <math>z</math> plane with
-levels <math>p=\Re(f)=\mathrm{const}</math> and
-levels <math>q=\Im(f)=\mathrm{const}</math>.
-Levels <math>p=-4,-3,-2,-1,0,1,2,3,4</math> are shown with thick black lines.
-The intermediate levels for <math>p<0 </math> are shown with thin red lines.
-The intermediate levels for <math>p>0 </math> are shown with thin blue lines.
-Levels <math>q=-4,-3,-2,-1</math> are shown with thick red lines.
-Levels <math>q=-1,2,3,4</math> are shown with thick blue  lines.
-The intermediate levels <math>q=\mathrm{const}</math> are shown with thin green lines.
-|author       = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
-|date-created = 2009 July 29
-|pub-country  = Japan
-|notes        = This image is expected to be used in the article [[superfunction]]
-|versions     = [[image:QfacQexp.jpg|100px]]
-}}
-== Licensing/Copyright status ==
-{{CC|by|3.0}}