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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Pentation from three fixed points
When b is a real number non inclusively between 1 and $e^{e^{-1}}$, $b^x$ has two real fixed points.
If b is increased to $e^{e^{-1}}$, the two real fixed points combine into one real fixed point. If ...
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How to directly approximate $f^n(z)$ for any real number $n$ where $f(z) = z^2 + c$
A while ago I came across this video, which shows how you can synthesize audio from the Mandelbrot set by calculating each value of $f^n(c)$, where $f^n(z) = f(f^{n-1}(z))$, $f(z) = z^2 + c$, and $c$ ...
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Find the curve of the iterated symmedian points
The symmedian point of the triangle with vertices $A=(-1,0),B=(1,0),C=(x,y)$ is $$F(x,y)=\left(\frac{4 x}{x^2+y^2+3},\frac{2 y}{x^2+y^2+3}\right)$$
By this question $F$ maps $\mathbb{R}^2$ onto an ...
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Exercise 3.39 in Function theory of one complex variables
${39.}$ Let $\varphi:D(0,1)\to D(0,1)$ be given by $\varphi(z)=z+a_2z^2+\ldots$. Define
$$\begin{align}
\varphi_{1}\left(z\right) &= \varphi\left(z\right), \\
\varphi_{2}\left(z\right) &=\...
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functional iteration and convolution [closed]
Question
$$ H(u,t)= u^{-1} (u X-1+e^{-uX }) $$
$$ H_T(u)=sup H(u,t) $$
$$H_T(u e^{-a v} ) <H_T(u) $$
$$ H_T(u)< u c^{-2} +2c_1 A(c_1 u) + H_T(u e^{-a v})$$
the author iterates the above equation ...
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Is there a way to compute oscillating iterated functions?
I've looked into iterated functions for a bit more than a year (especially thanks to tetration), but there's still things I do not quite know about them, especially when online searching isn't really ...
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Tetration Power Series
While reading through the Citizendium article on tetration, iterated exponentation, I came across a power series approximate of tetration. The article said that it got the series coefficients from ...
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What are all the tetration extension methods?
Tetration is the next step in our regular operations. Addition, multiplication, exponentiation, tetration.
It is constructed by repetitive exponentiations.
"$a$ tetration $b$" is written $^{...
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Fractional iteration of the exponential map z <- exp(lambda (z - w))
I want to use this map as a sort of chaotic oscillator for audio, where lambda and w are widgets you can control from something like a touch surface in real time. The map is from C to C, and lambda, z,...
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I want to make some curves from 0 to 1 using iterations
I want to make a few curves from 0 to 1 with a different growth acceleration.
So I tried this:
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Building Fractional Frequencies From Integer Frequencies
Given the ability to build any trigonometric polynomial of integer degrees:
$$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 ...
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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 ...
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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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