Difference between revisions of "File:Ernst schroederFragment.jpg"

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m (Format; add equations, links and the Keywords)
 
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{{oq|Ernst_schroederFragment.jp}}
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[[Ernst Schroeder]].
  
[[Ernst Schroeder]].
   
 
Fragment from image http://mizugadro.mydns.jp/t/index.php/File:Ernst_schroeder.jpg
  
Fragment from image http://mizugadro.mydns.jp/t/index.php/File:Ernst_schroeder.jpg
   
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Portrait of the german logician and mathematician [[Ernst Schroeder]] ([[Ernst Schröder]]). The photo was taken between 1890 and 1902.
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Portrait of the German logician and mathematician [[Ernst Schroeder]] ([[Ernst Schröder]]). The photo was taken between 1890 and 1902.
 
<ref>
  
<ref>
 
https://en.wikipedia.org/wiki/Ernst_Schröder
  
https://en.wikipedia.org/wiki/Ernst_Schröder
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Original filename: https://upload.wikimedia.org/wikipedia/commons/3/34/Ernst_schroeder.jpg
  
Original filename: https://upload.wikimedia.org/wikipedia/commons/3/34/Ernst_schroeder.jpg
   
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This image is used as Fig.6.3 at page 67 of book
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«[[Superfunctions]]»
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<ref>https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics - 2020/7/28
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</ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]].
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</ref><br>
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in order to attribute the [[Schroeder equation]]:<br>
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In [[Regular iteration]], the [[Schroeder equation]] appears as analogy of the [[Transfer equation]]; the [[Schroeder function]]
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appears as an analogy of the [[Superfunction]].<br>
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For trunsfer function \(T\) with [[fixed point]] zero
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(id est, \(T(0)\!=\!0\)),
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the pair ([[Schroederfunction]] \(S\), [[ArcSchroeder]] \(S^{-1}\)) allows to express iterates of \(T\) in the similar way, as it can be done with pair ([[Superfunction]], [[Abelfunction]]):
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  +
\[
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T^n(z)=S(n\ (S^{-1}(z))
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\]
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In this expression, number \(n\) of iterate has no need to be integer.
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The similar expression through
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the [[Superfunction]] \(F\) and
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the [[Abelfunction]] \(G\!=\!F^{-1}\) is
  +
  +
\[
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T^n(z)=F(n+G(z))
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\]
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==References==
  
==References==
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−
<references/>
  
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{{ref}}
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{{fer}}
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==Keywords==
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<b>«[[Ernst Schroeder]]»</b>,
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«[[Iterate]]»,
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«[[Regular iteration]]»,
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<b>«[[Schroeder equation]]»</b>,
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<b>«[[Schroederfunction]]»</b>,
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«[[Superfunctions]]»,
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«[[Transfer function]]»,
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«[[Transferfunction]]»,
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«[[Суперфункции]]»,
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<b>«[[Уравнение Шредера]]»</b>,
   
 
[[Category:Book]]
  
[[Category:Book]]
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[[Category:BookPhoto]]
 
[[Category:Ernst Schroeder]]
  
[[Category:Ernst Schroeder]]
 
[[Category:Fragment]]
  
[[Category:Fragment]]
 
[[Category:History]]
  
[[Category:History]]
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[[Category:Regular iteration]]
 
[[Category:Wikimedia]]
  
[[Category:Wikimedia]]

Latest revision as of 14:18, 20 August 2025


Ernst Schroeder.

Fragment from image http://mizugadro.mydns.jp/t/index.php/File:Ernst_schroeder.jpg

Portrait of the German logician and mathematician Ernst Schroeder (Ernst Schröder). The photo was taken between 1890 and 1902. [1]

Original filename: https://upload.wikimedia.org/wikipedia/commons/3/34/Ernst_schroeder.jpg

This image is used as Fig.6.3 at page 67 of book «Superfunctions» [2][3]
in order to attribute the Schroeder equation:
In Regular iteration, the Schroeder equation appears as analogy of the Transfer equation; the Schroeder function appears as an analogy of the Superfunction.
For trunsfer function \(T\) with fixed point zero (id est, \(T(0)\!=\!0\)), the pair (Schroederfunction \(S\), ArcSchroeder \(S^{-1}\)) allows to express iterates of \(T\) in the similar way, as it can be done with pair (Superfunction, Abelfunction):

\[ T^n(z)=S(n\ (S^{-1}(z)) \]

In this expression, number \(n\) of iterate has no need to be integer.

The similar expression through the Superfunction \(F\) and the Abelfunction \(G\!=\!F^{-1}\) is

\[ T^n(z)=F(n+G(z)) \]


References

  1. ↑ https://en.wikipedia.org/wiki/Ernst_Schröder
  2. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  3. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

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