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