Difference between revisions of "Iterate of linear fraction"

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Iterate of linear fraction (or iteration of linnet friaciton) refers to function
 
Iterate of linear fraction (or iteration of linnet friaciton) refers to function
−
$$g(z)=\frac{u+vz}{w+z}$$
 +
$g(z)=\frac{u+vz}{w+z}$
   
 
[[Iterate]] of a [[linear fraction]] can be expressed with also some linear fraction. This article describes this expression.
  
[[Iterate]] of a [[linear fraction]] can be expressed with also some linear fraction. This article describes this expression.
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==Special case==
  
==Special case==
 
First, consider the special case when the iterated function has only single parameter. Let
  
First, consider the special case when the iterated function has only single parameter. Let
−
$$f(z)=\frac{z}{c+z}$$
 +
$f(z)=\frac{z}{c+z}$
 
where $c\ne 0$ is constant; for example, the [[real number|real]] or [[complex number]].
 
where $c\ne 0$ is constant; for example, the [[real number|real]] or [[complex number]].
   
 
Iterate of this function can be denoted with
  
Iterate of this function can be denoted with
−
$$\Phi(n,z)=f^n(z)$$
 +
$\Phi(n,z)=f^n(z)$
   
 
Function $F$ should satisfy the following equations:
  
Function $F$ should satisfy the following equations:
−
$$\Phi(1,z)=f(z)$$
 +
$\Phi(1,z)=f(z)$
−
$${hi(n+1,z)=f(F(z))$$
 +
${hi(n+1,z)=f(F(z))$
 
For $c\ne 1$, the simple solution can be written as follows:
  
For $c\ne 1$, the simple solution can be written as follows:
−
$$\Phi(n,z)=\frac{z}{c^n+ \frac{1-c^n}{1-c} z }$$
 +
$\Phi(n,z)=\frac{z}{c^n+ \frac{1-c^n}{1-c} z }$
   
 
==Generalization==
  
==Generalization==
   
 
Consider linear function $P$, let
  
Consider linear function $P$, let
−
$$P(z)=a + b z$$
 +
$P(z)=a + b z$
   
 
The inverse function can be written as follows:
  
The inverse function can be written as follows:
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One can easy check that $P(Q(z))=Q(P(z))=z$. Let
  
One can easy check that $P(Q(z))=Q(P(z))=z$. Let
−
$$g(z)=P(f(Q(z)))=
 +
$g(z)=P(f(Q(z)))=
 
\frac{abc - ab - a^2 ~+~ (a\!+\!b)z}
  
\frac{abc - ab - a^2 ~+~ (a\!+\!b)z}
−
{bc-a+z}$$
 +
{bc-a+z}$
   
 
This agree with equation (1) at
  
This agree with equation (1) at
−
$$u=abc-ab-a^2$$
 +
$u=abc-ab-a^2$
−
$$v=a+b$$
 +
$v=a+b$
−
$$w=bc-a$$
 +
$w=bc-a$
   
 
It is possible to express $a,b,c$ in terms of $u,v,w$:
  
It is possible to express $a,b,c$ in terms of $u,v,w$:
−
$$a=\frac{v-w-\Gamma}{2}$$
 +
$a=\frac{v-w-\Gamma}{2}$
−
$$b=\frac{v+w+\Gamma}{2}$$
 +
$b=\frac{v+w+\Gamma}{2}$
−
$$c=\frac{-2 u - v^2 - w^2 +(v+w)\Gamma}
 +
$c=\frac{-2 u - v^2 - w^2 +(v+w)\Gamma}
−
{2(u-vw)}$$
 +
{2(u-vw)}$
 
where
 
where
−
$$Gamma=\pm \sqrt{4 u + v^2 - 2 v w + w^2}$$
 +
$Gamma=\pm \sqrt{4 u + v^2 - 2 v w + w^2}$
   
 
==Non-integer iterate==
  
==Non-integer iterate==
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As function $g$ is expressed through [[conjugation]] of function $f$,
  
As function $g$ is expressed through [[conjugation]] of function $f$,
 
the iteration can be expressed as follows:
  
the iteration can be expressed as follows:
−
$$g^n(z)=P(f^n*Q(z))$$
 +
$g^n(z)=P(f^n*Q(z))$
 
all functions P,f^n and $Q$ are defined above, and $f^n$ is expressed in a way, that does not require the number $n$ of iteration to be integer.
  
all functions P,f^n and $Q$ are defined above, and $f^n$ is expressed in a way, that does not require the number $n$ of iteration to be integer.
   
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The nth iteration of function $f$ can be used to express the [[superfunction]] for the linear fraction; if $f$ is declared as [[transfer function]], then, for some constant $t$, the superfunciton $F$ can be written as follows:
  
The nth iteration of function $f$ can be used to express the [[superfunction]] for the linear fraction; if $f$ is declared as [[transfer function]], then, for some constant $t$, the superfunciton $F$ can be written as follows:
   
−
$$ F(z)=f^z(t)$$
 +
$ F(z)=f^z(t)$
   
 
and the interse function $G=F^-1$ appears as the [[Abel function]].
  
and the interse function $G=F^-1$ appears as the [[Abel function]].

Revision as of 15:42, 29 July 2013

Iterate of linear fraction (or iteration of linnet friaciton) refers to function 

$g(z)=\frac{u+vz}{w+z}$

Iterate of a linear fraction can be expressed with also some linear fraction. This article describes this expression.

Special case

First, consider the special case when the iterated function has only single parameter. Let 

$f(z)=\frac{z}{c+z}$

where $c\ne 0$ is constant; for example, the real or complex number.

Iterate of this function can be denoted with 

$\Phi(n,z)=f^n(z)$

Function $F$ should satisfy the following equations: 

$\Phi(1,z)=f(z)$
${hi(n+1,z)=f(F(z))$

For $c\ne 1$, the simple solution can be written as follows: 

$\Phi(n,z)=\frac{z}{c^n+ \frac{1-c^n}{1-c} z }$

Generalization

Consider linear function $P$, let 

$P(z)=a + b z$

The inverse function can be written as follows: 

$$Q(z)=P^{-1}(z)=\frac{x-a}{b}$$

One can easy check that $P(Q(z))=Q(P(z))=z$. Let 

$g(z)=P(f(Q(z)))=

\frac{abc - ab - a^2 ~+~ (a\!+\!b)z} 

    {bc-a+z}$

This agree with equation (1) at 

 $u=abc-ab-a^2$
 $v=a+b$
$w=bc-a$

It is possible to express $a,b,c$ in terms of $u,v,w$: 

$a=\frac{v-w-\Gamma}{2}$
$b=\frac{v+w+\Gamma}{2}$
$c=\frac{-2 u - v^2 - w^2 +(v+w)\Gamma}
         {2(u-vw)}$

where 

$Gamma=\pm \sqrt{4 u + v^2 - 2 v w + w^2}$

Non-integer iterate

As function $g$ is expressed through conjugation of function $f$, the iteration can be expressed as follows: 

$g^n(z)=P(f^n*Q(z))$

all functions P,f^n and $Q$ are defined above, and $f^n$ is expressed in a way, that does not require the number $n$ of iteration to be integer.

In such a way, the linear fraction can be iterated any number of times, the halfiteration and the complex iterations are straightforward. In Mathematica, this could be expressed with the built-in function Nest, but, up to year 2013, the software does not yet allow to use it for non-integer number of iterate; if the number of iterations cannot be expressed with some positive integer constant, Mathematica generates the error message instead of to evaluate the expression.

Superfunction

The nth iteration of function $f$ can be used to express the superfunction for the linear fraction; if $f$ is declared as transfer function, then, for some constant $t$, the superfunciton $F$ can be written as follows: 

$ F(z)=f^z(t)$

and the interse function $G=F^-1$ appears as the Abel function.

Linear fraction is one of transfer functions, fot which the superfunction and the Abel function can be expressed in terms of elementary functions.

References

http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188.

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
D. Kouznetsov. Solution of F(x+1)=exp(F(x)) in complex z-plane. 78, (2009), 1647-1670

http://www.ils.uec.ac.jp.jp/~dima/PAPERS/2009vladie.pdf (English)
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf (English)
http://mizugadro.mydns.jp/PAPERS/2009vladir.pdf (Russian version)
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

http://reference.wolfram.com/mathematica/ref/Nest.html Nest, Wolfram Mathematica 9 Documentation center, 2013.

Keywords

Abel function, Iteration, Superfunction, Tetration,

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