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==Redirected page==
[[Function Iteration]] was redirected to [[Function composition]]. Should this article redirect there as well (after any necessary merging)? - [[User:Dcljr|dcljr]] <small>([[User talk:Dcljr|talk]])</small> 18:58, 11 November 2005 (UTC)
:No. Please note '''this''' comment should have been placed at the bottom of the talk page, in the first place. [[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 18:57, 28 December 2015 (UTC)
==Merge with Recurrence relation?==
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Surely the expressions can be by far non rigurous, but can be someone to look my notes on [http://www.iw-net.org/index.php?title=Theory_of_continuous_composition] and improve that or comment about? <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.25.164.213|80.25.164.213]] ([[User talk:80.25.164.213|talk]]) 20:06, 20 July 2012 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
:I sense you are in the wrong article, except perhaps for the brief summary remarks of the "Conjugacy" section. You are replicating, in somewhat idiosyncratic language, the continuous iteration orbit theory of [[Schröder's equation]]. Possibly {{
== My little contribution ==
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: Of course your name is better, but handling dead links could be a nightmare. And links is what WP is all about. As a stopgap, you should make a redirect.
: By the way, "splinter" is a standard term, used interchangeably with "Picard sequence" in less parochial corners of the broad and disparate user community, and is borrowed from recursive function theory, ''Ullian, J., Splinters of Recursive Functions, The Journal of Symbolic Logic, Vol. 25, N. 1, March, 1960, pp. 33 - 38''.[[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 11:32, 18 February 2015 (UTC)
::A redirect [[Function iteration]]-->[[Iterated function]] is present since 2006. Perhaps the lead should be rephrased to cope with both titles. What about:
:::In [[mathematics]], '''function iteration''' is the process of [[function composition|composing]] a function ''f'': ''X'' → ''X'' (that is, a [[function (mathematics)|function]] from some [[set (mathematics)|set]] ''X'' to itself) with itself a certain number ''n'' of times. In this process, starting from some initial element of ''X'', the output of ''f'' is fed again into ''f'' as input, and this process is repeated. The result of function composition is again a function ''f''<sup>''n''</sup>: ''X'' → ''X'', it is called an '''iterated function'''.
::or something similar? The sentence ''"The process of repeatedly applying the same function is called [[iteration]]."'' could be moved down e.g. to the "Definition" section, and adapted to e.g. ''"Function iteration is a particular example of [[iteration]] in general."'' if it is considered worth to be kept at all. In contrast, the possibility of fractional iterates ''should'' be mentioned already in the lead, I think. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 17:31, 18 February 2015 (UTC)
== Slight Ambiguity ==
When the article says, "Note: these two special cases of ax2 + bx + c are the only cases that have a closed-form solution. Choosing b = 2 = –a and b = 4 = –a, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table." Does it mean that we only know of two closed form solutions, or has it been proven that there are no others? <small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/75.243.141.93|75.243.141.93]] ([[User talk:75.243.141.93|talk]]) 21:20, 4 October 2015 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
:Well, a "proof" for the absence of a closed form solution might be hard to conceive, mostly because "closed form" is a bit of a subjective label: it is predicated on agreement of what the "closed form" solutions of [[Schröder's equation]] may include. You may see, though, in conventional language, from the original Schroeder paper of 1870 that they really ''have'' to be just these two for the [[logistic map]], and the Katsura and Fukuda paper pushes the envelope, but ''only'' comes up with trivial changes of variables. If you thought you found new ones which do not rely on implicitly ''defining'' new functions solving the Schroeder equation, you might well suggest them on this talk-page. [[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 22:50, 4 October 2015 (UTC)
::ik this is very late but what does the Katsura and Fukuda say. [[Special:Contributions/100.2.153.196|100.2.153.196]] ([[User talk:100.2.153.196|talk]]) 02:00, 29 November 2023 (UTC)
:::also is there any way to access these papers without paying [[Special:Contributions/100.2.153.196|100.2.153.196]] ([[User talk:100.2.153.196|talk]]) 02:13, 29 November 2023 (UTC)
== Complex Iteration ==
I've discovered that not only fractional iteration is possible, but also complex iteration. It's not as intuitive as fractional iteration, but it does make sense once you have a new model for what iteration means. Probably the simplest explanation is with orbitals in the complex plain: iterating to a real values generates the complex orbitals as paths. Iterating to imaginary powers, meanwhile, generates paths perpendicular to the orbitals at every point. You need the entire vector field of orbitals to know the new path. For example, iterating a linear function (multiplying by a constant) creates orbitals which are rays. Iterating to imaginary powers creates paths with are concentric circles. After that, complex iteration can be done by composing real and imaginary values. For a more precise derivation, I had to invent a new type of derivative, it looks at the instantaneous change in value at a point as the function iterates. The formula for this derivative is:
f*(x) = lim n->0 (f<sup>n</sup>(x) - x)/n
Which is equivalent to the derivative of f's Abel function. The key about this derivative is it has the property that f<sup>n</sup>* = f x n . This enables iteration to imaginary powers to be calculated directly. Ultimately I think it is equivalent to taking the analytic extension of the Abel function, but it's a bit more intuitive than Taylor series witchcraft. I've done the calculations to arrive at Euler's identity using this method rather than the typical one, and graphically it makes much more sense: once it's understood how complex iteration works, it's just geometry to find the imaginary value e must be raised to in order to complete the arc from 1 to -1. <small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/216.49.181.128|216.49.181.128]] ([[User talk:216.49.181.128|talk]]) 15:34, 17 December 2015 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
::Please heed & comply with above rubric, [[Wikipedia:What Wikipedia is not#FORUM]], [[Wikipedia:No original research]]. [[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 19:17, 17 December 2015 (UTC)
== Question:How did the editor get away with treating "n" like a regular power??? ==
In one of the subsections, it says "When n is not an integer, make use of the power formula y n = exp(n ln(y))". This will not work at all and makes no sense, n is not a power, it is the number of iterations. That line should be deleted.
There's no reply button, so I'll just have to edit here to say that I don't see anything in the Curtright, T.L. Evolution surfaces and Schröder functional methods. That article does not address what I asked, and instead of using facts and evidence, someone trolled by editing my question to say the article did, so I am reporting them for a moderator to look into.
I have not seen anyone actually bring up any logical contention with this change, so in two days (from my time zone) I will delete that sentence. Posted by [[User:Leakdope]] without signature, 12/2018.
````Leakdope```` <!-- Template:Unsigned --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Leakdope|Leakdope]] ([[User talk:Leakdope#top|talk]] • [[Special:Contributions/Leakdope|contribs]]) 04:09, 12 December 2018 (UTC)</small> <!--Autosigned by SineBot-->
:Please '''''sign''''' your postings with four tildes, ~ , as instructed by your edit session. [[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 15:13, 11 December 2018 (UTC)
== Placing the [[Iterated function]] and [[Tetration]] pages on more solid mathematical ground ==
Please see my comments at [[Talk:Tetration#Moving_towards_a_verifiable_article]] which is relevant to both articles.
[[User:Daniel Geisler|Daniel Geisler]] ([[User talk:Daniel Geisler|talk]]) 18:53, 30 April 2019 (UTC)
:Isn't this a bit scattershot? This article here is reasonably well sourced, and anything and all connections to tetration may reasonably be severed, probably usefully, no? Fractional iteration is a mature functional conjugacy field and amply sourced and documented. If you wished to delete the generic unnumbered "example" 6.2 sourced to the Tetration site, you should be welcome, after proposing specific defensible deletions: proceed! [[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 19:34, 30 April 2019 (UTC)
Sorry, but I strongly disagree with the statement that fractional iteration is a mature functional conjugacy field. Papers are being published on extending tetration to the real and complex numbers solely on Abel's Functional Equation which is wrong because it is only valid for <math>f(z)</math> once the coordinate system is shifted to a fixed point <math>f(0)=0</math> then <math>f'(0)=1</math> must be true. I am in the process of writing something up to send to these authors and the editors of the journals they published in.
If you review the older material from the beginnings of the [https://math.eretrandre.org/tetrationforum/index.php Tetration Forum] you will see them arguing that fixed points were not important, while my work starts with fixed points. So this work is inconsistent with Schroeder's ''Über iterirte Functionen'', considered the first paper on dynamics.
Even though I try and track people and papers, both for myself and the community of mathematicians I communicate with. Maybe our different views of functional conjugacy are based on you being connected to people and papers I am unaware of. I'd appreciate any information you wish to share, either publically or privately. Thank you.
[[User:Daniel Geisler|Daniel Geisler]] ([[User talk:Daniel Geisler|talk]]) 20:45, 1 May 2019 (UTC)
:Sorry this is not a forum. A combination of Schroeder's, Abel's, or Boettcher's equations, all linked to cover each other's blind spots have been used for over a century, and this is the somewhat dull attested work covered here. Again, this is not a forum, and not a Tetration venue. [[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 21:23, 1 May 2019 (UTC)
I'm sorry then, I see that we will not be agreeing. This is not an appropriate place for me to benefit people. I'm fine with letting history judge the merits of our work here. So best wishes to the folks here.
[[User:Daniel Geisler|Daniel Geisler]] ([[User talk:Daniel Geisler|talk]]) 07:49, 2 May 2019 (UTC)
== The article would be greatly improved ==
by the omission of the top illustration, which is overwhelmingly confusing.
It is confusing simply because it tries to convey about ten times as much information as one illustration can convey.
Often — and in this case — '''less is more'''.
'''If''' the illustration were replaced by a far simpler one, having almost no text, that would be a good thing. [[Special:Contributions/2601:200:C000:1A0:5FB:D9A8:4BAF:D605|2601:200:C000:1A0:5FB:D9A8:4BAF:D605]] ([[User talk:2601:200:C000:1A0:5FB:D9A8:4BAF:D605|talk]]) 19:34, 24 November 2022 (UTC)
== Bad article ==
This article contains multiple passages with bad, confusing, and/or misleading writing. The top illustration is at least as confusing as any other bad illustration in Wikipedia, but even more so. Significant important relevant topics, like Koenigs's theorem as well as the central topic of embedding certain functions into a continuous flow, are left entirely unmentioned. And the passage about iterating f(x) = (√2)<sup>x</sup> is a train wreck. - 2601:200:c082:2ea0:a874:6184:ede8:53c1 11:56, 21 May 2023
:You are invited to flag each passage with an appropriate template, see [[Template:Inline cleanup tags]] for an overview, or even to improve the text yourself, cf. [[WP:Bold]]. Also, you may replace the top illustration by a better one, and suggest sections about unmentioned topics. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 12:10, 21 May 2023 (UTC)
== actual half iterate of x ==
if you use the f^0(x) axiom,as well as the exponentioation property of iterated functions you can conclude that the half itarate of x aka the identy map is itself. [[Special:Contributions/100.2.153.196|100.2.153.196]] ([[User talk:100.2.153.196|talk]]) 01:24, 20 November 2023 (UTC)
:I guess there is an error in your proof. Can you give it in detail? - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 11:00, 20 November 2023 (UTC)
::1. g(x)=f^0(x)=x
::2.g^n(x)=nth iterate of f^0(x)=f^0n(x)=f^0(x)=x [[Special:Contributions/168.100.171.2|168.100.171.2]] ([[User talk:168.100.171.2|talk]]) 14:54, 20 November 2023 (UTC)
:::What are you going to prove, what is f, what is g? Why should 1. hold? Can you express 2. in a formal correct way, without English text appearing in expressions? - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 08:18, 21 November 2023 (UTC)
::::1. holds as f^0(x) always equals x its a given in the deinition of an iterated function
::::2. i had no notation that could do that but up, but i relized the wikiedia article has some
::::3. revised proof
::::g(x)=x=f^0(x)
::::g^n(x)=(f^0)^n(x)=f^0(x)=x [[Special:Contributions/100.2.153.196|100.2.153.196]] ([[User talk:100.2.153.196|talk]]) 03:34, 22 November 2023 (UTC)
:::::minor mistake instaid of jumping from f^0^n(x) to f^0(x) i should go to f^0n(x) [[Special:Contributions/100.2.153.196|100.2.153.196]] ([[User talk:100.2.153.196|talk]]) 03:35, 22 November 2023 (UTC)
::::::Again: If anybody is supposed to follow your proof, you should bring it into a strictly mathematicel form, as found in typical math textbooks. You should initially state your claim. In the following proof, every step should be justified, as an assumption, from a definition, or as an inference of earlier results. The claimed property should be the last step. Like e.g. "{{tq|1=I claim that for every ... we have ... . To prove it, let ... be ... . Then, we have ... = ... = ... by ... and ..., respectively. Hence, ... by ... . etc. ... Hence, we are done.}}" - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 10:13, 22 November 2023 (UTC)
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