Questions tagged [fractional-iteration]

The study of fractional self-iterations of a map. A basic example is the analysis of functional square roots of a map $g$, i.e. solutions $f$ to the functional equation $f\circ f=g$ are functional square roots and solutions to $f^n=g$ are functional nnth roots.

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Is there a way to compute oscillating iterated functions?

I've looked into iterated functions for a bit more than a year (especially thanks to tetration), but there's still things I do not quite know about them, especially when online searching isn't really ...
Pierre Carlier's user avatar
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Tetration Power Series

While reading through the Citizendium article on tetration, the first hyper-operation above exponentiation, I came across a power series approximate of tetration. The article said that it got the ...
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What are all the tetration extension methods?

Tetration is the next step in our regular operations. Addition, multiplication, exponentiation, tetration. It is constructed by repetitive exponentiations. "$a$ tetration $b$" is written $^{...
Pierre Carlier's user avatar
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Fractional iteration of the exponential map z <- exp(lambda (z - w))

I want to use this map as a sort of chaotic oscillator for audio, where lambda and w are widgets you can control from something like a touch surface in real time. The map is from C to C, and lambda, z,...
Emanuel Landeholm's user avatar
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I want to make some curves from 0 to 1 using iterations

I want to make a few curves from 0 to 1 with a different growth acceleration. So I tried this: ...
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Building Fractional Frequencies From Integer Frequencies

Given the ability to build any trigonometric polynomial of integer degrees: $$T_d(\theta) = \sum_{n=-d}^d c_n e^{in\theta}$$ I wish to construct, or technically approximate: $$e^{it\theta} \>\>\&...
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How to derive a recursive formula from the following formula

How to derive a recursive formula from the following formula, $$ u_{n}=a_{n-1}u_{0}+\sum_{k=1}^{n-1}(a_{n-1-k}-a_{n-k})u_{k}+\Gamma(2-\alpha)h^{\alpha}f(t_{n},u_{n})? $$ P.S.: Consider the following ...
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Solve $T(n)=T(\frac{n}{a})+T(\frac{n}{b})+n^{k}$

is there known way to solve iterative equasion to direct one: $T(n)=T(\frac{n}{a})+T(\frac{n}{b})+n^{k}$ if the starting condition like $T(n)=c$ is known or maybe you can invent one? thanks for ...
user184868's user avatar
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How can I create a fomula to calculate iterations between 2 numbers where on keep reducing at each iteration?

I am a software Engineer and I've run into an issue and I need to generate a kind of formula that will help me calculate iterations. I thought of asking this question on stackoverflow but what I need ...
Henry Obiaraije's user avatar
2 votes
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Generalized nested sine function [closed]

As sine functions are nested more and more in manner shown below, the shape of the function approaches that of a square wave. \begin{align} f^1(x)&=\sin(x)\\ f^2(x)&=\sin(\,\sin(x)\,)\\ f^3(x)&...
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What are suitable functions for series expansions of fractional composition roots of functions?

Lately I have been investigating numerical methods to approximate fractional composition roots of functions. My intuition tells me that power series expansions will not be very suitable in ...
mathreadler's user avatar
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Fourier series of iterated sin / Diagonalization of an infinite matrix of Bessel functions

We define the iterated sine function as : $$ \sin^n(x) = \sin(\sin(.... \sin(x)))\:\:n\:\text{times.} $$ We know the "Frequency Modulation" formula based on Bessel functions :$$ \sin( p\, \...
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Iteration of logarithm

Searching "Iterated logarithm" seems to give results only about the function $\log*(x)$. I would like to read about the properties of functions such as $$\begin{align} \log_{1}(x)&=\log(...
Michael Riberdy's user avatar
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Construct a function $f$ that $f(f(z))=-z(1-z)$ without using any information linked to $f(f(2))=2$

Consider the analytic function $g(z)=-z(1-z)$ which is a generator of Logistic Sequence with multiplier $-1$, having 2 global fixed point, $g(0)=0$ and $g(2)=2$. From classical dynamics, whenever a ...
Leo Warvin Peng's user avatar
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How to generate $O(u_1^6+u_4^4)$ series

In the paper FRACTIONAL ITERATION OF SERIES AND TRANSSERIES by G. A. EDGAR, the solution to $f^s(x)=x^2+c$ is given: $$(19)\;\;\; M^{[s]} = \exp\circ M_1^{[s]}\circ\log = x^{2^s}\left(1+2^{-1+s}c\...
Max's user avatar
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Is there an method similar to FFT for Fractionally Iterated Fourier Transforms?

FFT is one of the 20th Century's greatest inventions, running as $O(n \log(n))$ rather than as $O(n^2)$ as a simple implementation of a discrete Fourier transform would. But what about half-order ...
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About iteration Matrix

Note that the size of the linear system (number of equations and unknowns) corresponding to the image blurring process is proportional to the number of pixels in the image. Therefore, even for a ...
Jeremy's user avatar
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Does this nested log function exist?

I was wondering if the following function exists: lets say you have $$f^1(x) = \ln(x)$$ $$f^2(x) = \ln(\ln(x))$$ $$f^3(x) = \ln(\ln(\ln(x)))$$ $$f^4(x) = \ln(\ln(\ln(\ln(x))))$$ and so forth is there ...
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Discrete fractional iteration?

I’m teaching a course in discrete mathematics and was exploring (purely for my own interest) whether there were any functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(f(n)) = n + 1$. I then ...
templatetypedef's user avatar
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Are there any relations between fractional calculus and fractional iteration?

I wonder how the field of fractional calculus and the study of fractional iterates of functions relate to one another. I'm interested in cases in which techniques from the former field can be used to ...
Max Muller's user avatar
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A general theory of fractional operations?

This post is motivated by observing that certain operators have "fractional analogs". For example, let $f: R\to R$ be a differentiable function. The differential operator $D^{1}(f)$ produces ...
Albert Zevelev's user avatar
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Numerically find the half-iterate of a quadratic

Let's say we have a function $f(x)=n+mx+lx^2$, and want to find another function $h(x) = a+bx+cx^2 $ such that $h(h(x)) = f(x)$. Here's what you get when composing h on itself: $$h(h(x)) = a+ab+b^{2}...
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What is known about the functional square root of the Riemann Zeta function?

Let us consider the Riemann zeta function $\zeta(s)$ for $Re(s) > 1$: $$\zeta(s) := \sum_{n=1}^{\infty} \frac{1}{n^{s}} .$$ I wonder what is known about the functional square root(s) of the ...
Max Muller's user avatar
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nonlinear system of equations solved by iteration method does not converge

I've modeled an electrical machine by means of a magnetic equivalent circuit. I've reached to a system of equations like this $ [f(F_N)]_{N\times1}^q=[P_N]_{N\times N}^q*[F_N]_{N\times1}^q-[\phi_N]_{N\...
Scrat68's user avatar
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A method to iterate the exponential function a non-integer number of times?

Notation We employ the following notation: $$ a_1 = e^{x} $$ $$ a_2 = e^{e^{x}} $$ $$ a_3 = e^{e^{e^{x}}} $$ $$ a_n = e^{\vdots^{e^{x}}}$$ We also use define $c$ by: $$ e^c =c$$ Motivation ...
More Anonymous's user avatar
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Maximally connected subset is surjective

I got a question on this proof I read in a paper. Let $f(z)=\frac{e^z}{z+1}$. We know that $-1$ is the only pole of the function. Now we take $L_0=[-1,\infty) \cup \infty$(only real axis) and $L_{-...
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Uniquely extended fractional iterations of $\exp$

Let us define the following basic conditions for an iterated exponential function: $$\exp^1(x)=e^x\tag{$\forall x$}$$ $$\exp^{a+b}(x)=\exp^a(\exp^b(x))\tag{$\forall a,b,x$}$$ I then pondered what ...
Simply Beautiful Art's user avatar
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Iterative functional equation $f \circ f = f$ [duplicate]

I tried but failed to find some references about the following iterative functional equation: find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $f \circ f = f$, that is $f(f(x))=f(x)$ ...
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Functions $f$ that $f(f(x))=x$, but $f:S^1\to S^1$

Background Denote $e_A$ the identity map from $A$ to itself. Questions such like solving $f$ in the functional equation $f\circ f=e_\mathbb{R}$ or $f\circ f=e_{\mathbb{R}\setminus\{a_1,\ldots,a_n\}}$ ...
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A conjectured identity for the fractional iterates of $\sqrt{x}+2$

Deeply impressed by the Fractional iterates of $x²-2$, a problem by Ramanujan post one month ago, I decided to investigate a little by myself on the topic and discovered that using Carleman matrices ...
Thomas Baruchel's user avatar
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Real numbers arbitrary closeness (iterated fraction)

I was playing with some iterated functions/fractions when I stumbled across this one $$\frac{1+\frac{1+\frac{1+...}{1-...}}{1-\frac{1+...}{1-...}}}{1-\frac{1+\frac{1+...}{1-...}}{1-\frac{1+...}{1-...}}...
eagr's user avatar
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How can I create currency note count calculator if I have currencies which has decimals and total amount which also has decimals?

If i have currencies that are in decimals {0.7,0.8,1.4,0.5} and total amount which also has decimals (example: 1279.6) and I tell you that you need to make total within certain number of currency note ...
GirKarr's user avatar
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Fourier transform and fourth root

Given a well-behaved convex function $f(t):\mathbb{R}\to \mathbb{R}$, its Fourier transform (FT) $\hat{f}(\omega)=\mathcal{F}[f(t)](\omega)$ is positive (and decreasing) proof here. It follows that ...
Chaos's user avatar
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Find $f(x)$ satisfying $f(f(x))=x^x$

By inspection my attempts are always wrong. I really have no idea and given up. How to find $f(x)$ satisfying $f(f(x))=x^x$? My attempts: $f(x)=x^x$ $f(x)=x^{1/x}$ $f(x)=\frac{1}{x^x}$ My ...
D G's user avatar
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Is the Schröder Equation valid for higher dimensional iterated maps?

The Schröder's functional equation is the eigenfunction equation for the composition operator given as: $$ \psi \circ y (x) = s \cdot \psi(x) ~~~~~~~~~~~ (1) $$ The interesting bit about it (at ...
urquiza's user avatar
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1 vote
1 answer
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Is there any definition of the sum of $n$ summands, when $n$ is not a natural number? [duplicate]

I think, from the algebraic point of view, that this doesn't make sense, since the addition of $n$ terms is defined "inductively" from the binary operation of the addition. This even yields the ...
Manuel Sánchez's user avatar
13 votes
3 answers
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Find $f(f(\cdots f(x)))=p(x)$

$\newcommand{\nest}{\operatorname{nest}}$Let's define a function $\nest(f, x, k)$, which takes a function $f$, an input $x$, and a non-negative integer $k$, and calls $f$ on $x$ repeatedly ($k$ times)....
Antonio Perez's user avatar
4 votes
1 answer
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Functional Square Root of Digit Sum?

Define $\text{sdig}(n)$ to be the sum of the decimal digits of $n$, where $n$ is a positive integer. My question is as follows: Does there exist a function $h:\mathbb Z^+\mapsto\mathbb Z^+$ such ...
Franklin Pezzuti Dyer's user avatar
-1 votes
1 answer
285 views

Do two exponential spirals intersect?

I have lists of complex points: orbit of complex point z under quadratic function f(z) = z*z I know that lists are: z, z^2, z^4, z^8, ... (r,t), (r^2, 2*t), .....
Adam's user avatar
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2 votes
1 answer
351 views

Golden ratio - almost integer

Let, $a_1=1, b_1=3, c_1=3$ and $d_1=1$ Where $a_2=a_1+c_1,b_2=b_1+d_1, c_2=a_1+b_1+c_1 $ and $d_2=a_1+d_1$ and we defined $$a_n=a_{n-1}+c_{n-1}$$ $$b_n=b_{n-1}+d_{n-1}$$ $$c_n=a_{n-1}+b_{n-1}+c_{...
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2 votes
2 answers
158 views

Almost-half iterate of $x^2+1$

Because I can't find a function $h:\mathbb R\mapsto\mathbb R$ with the property $$h^{\circ 2}(x)=x^2+1$$ I'm looking for a function that almost has that property - that is, I would like to find a ...
Franklin Pezzuti Dyer's user avatar
5 votes
1 answer
131 views

If $q(x)=x^2+1$, does $q^{\circ 1/2}$ exist? [duplicate]

I've been doing a lot of research about functional half-iteration, and I posed the following question to myself: Consider the function $q:\mathbb R\mapsto\mathbb R$ defined as $$q(x)=x^2+1$$ ...
Franklin Pezzuti Dyer's user avatar
1 vote
1 answer
706 views

Functional Square Root of Piecewise Functions

Let $s:\mathbb R\to\mathbb R$ be the function defined as $$s(x)=x+(-1)^{\lfloor x\rfloor}$$ Find a function $t:\mathbb R\to \mathbb R$ such that $t^{\circ 2}=s$, or prove that no such function ...
Franklin Pezzuti Dyer's user avatar
0 votes
1 answer
105 views

Partial iteration and inverse iteration of $e^x$

Find the general $f^n(x)$ where $f^1(x)=e^x$ $f^{a}(f^{b}x)=f^{(a+b)}(x)$ where $n,x\in\mathbb R$ I'm fairly confident that no two $f^n(x)$ with different values of $n$ will intersect, since then we ...
Jacob Claassen's user avatar
21 votes
11 answers
6k views

half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and $x&...
Sheldon L's user avatar
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