Hyperoperations Wiki - User contributions [en]

Cauchy Method

2016-02-06T13:11:14Z

Kouznetsov: /* Description */

[[Cauchy Method]] of evaluation of [[superfunction]] refers to the real–holomorphic transfer function $T$ that has no real [[fixed point]], but has at least one pair of complex fixed points, $L$ and $L^*$.

The method is supposed to evaluate superfunction that approaches
$L$ at $\mathrm i \infty$
and approaches
$L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/\mathrm e)$.

and the applicability to other functions required additional investigation.

==Description==

The [[transfer equation]] for the superfunction $F$ of transfer function $T$
Can be written as follows:

$
F(z+1)=T(F(z))
$

Within some domain of values of $z$,
the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z))$

Then, for some strip in direction of imaginary axis,
the transfer equation can be written through the [[integral Cauchi]]

$\displaystyle
F(z)=\oint \frac{F(t)}{t-z} \mathrm d t
$

where the integration is performed along the boundary of the strip.
In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$;
for long "strip" this is supposed to give the precise approximation of the integral.
The width of the "strip" along the real part of $t$ should be equal to 2;
then, the transfer equation and its inverse allow to represent values
a the edges of the "strip" through the values along the line of the strip.
In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation;
that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.

==References==
<references/>

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. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

Cauchy Method

2016-02-06T13:10:59Z

Kouznetsov: /* Description */

[[Cauchy Method]] of evaluation of [[superfunction]] refers to the real–holomorphic transfer function $T$ that has no real [[fixed point]], but has at least one pair of complex fixed points, $L$ and $L^*$.

The method is supposed to evaluate superfunction that approaches
$L$ at $\mathrm i \infty$
and approaches
$L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/\mathrm e)$.

and the applicability to other functions required additional investigation.

==Description==

The [[transfer equation]] for the superfunction $F$ of transfer function $T$
Can be written as follows:

$
F(z+1)=T(F(z))
$

Within some domain of values of $z$,
the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z))$

Then, for some strip in direction of imaginary axis,
the transfer equation can be written through the [[integral Cauchi]]

$\displaystyle
F(z)=\oint \frac{F(t)}{t-z} \mathrm d t
$
where the integration is performed along the boundary of the strip.
In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$;
for long "strip" this is supposed to give the precise approximation of the integral.
The width of the "strip" along the real part of $t$ should be equal to 2;
then, the transfer equation and its inverse allow to represent values
a the edges of the "strip" through the values along the line of the strip.
In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation;
that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.

==References==
<references/>

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. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

Cauchy Method

2016-02-06T13:10:38Z

Kouznetsov: /* Description */

[[Cauchy Method]] of evaluation of [[superfunction]] refers to the real–holomorphic transfer function $T$ that has no real [[fixed point]], but has at least one pair of complex fixed points, $L$ and $L^*$.

The method is supposed to evaluate superfunction that approaches
$L$ at $\mathrm i \infty$
and approaches
$L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/\mathrm e)$.

and the applicability to other functions required additional investigation.

==Description==

The [[transfer equation]] for the superfunction $F$ of transfer function $T$
Can be written as follows:

$
F(z+1)=T(F(z))
$

Within some domain of values of $z$,
the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z))$

Then, for some strip in direction of imaginary axis,
the transfer equation can be written through the [[integral Cauchi]]

$
F(z)=\oint \frac{F(t)}{t-z} \mathrm d t
$
where the integration is performed along the boundary of the strip.
In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$;
for long "strip" this is supposed to give the precise approximation of the integral.
The width of the "strip" along the real part of $t$ should be equal to 2;
then, the transfer equation and its inverse allow to represent values
a the edges of the "strip" through the values along the line of the strip.
In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation;
that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.

==References==
<references/>

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. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

Cauchy Method

2016-02-06T13:10:17Z

Kouznetsov: /* Description */

[[Cauchy Method]] of evaluation of [[superfunction]] refers to the real–holomorphic transfer function $T$ that has no real [[fixed point]], but has at least one pair of complex fixed points, $L$ and $L^*$.

The method is supposed to evaluate superfunction that approaches
$L$ at $\mathrm i \infty$
and approaches
$L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/\mathrm e)$.

and the applicability to other functions required additional investigation.

==Description==

The [[transfer equation]] for the superfunction $F$ of transfer function $T$
Can be written as follows:

$
F(z+1)=T(F(z))
$

Within some domain of values of $z$,
the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z))$

Then, for some strip in direction of imaginary axis,
the transfer equation can be written through the [[integral Cauchi]]

$
F(z)=\oint \frac{F(t)}{t-z)} \mathrm d t
$
where the integration is performed along the boundary of the strip.
In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$;
for long "strip" this is supposed to give the precise approximation of the integral.
The width of the "strip" along the real part of $t$ should be equal to 2;
then, the transfer equation and its inverse allow to represent values
a the edges of the "strip" through the values along the line of the strip.
In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation;
that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.

==References==
<references/>

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. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

Cauchy Method

2016-02-06T13:09:50Z

Kouznetsov:

[[Cauchy Method]] of evaluation of [[superfunction]] refers to the real–holomorphic transfer function $T$ that has no real [[fixed point]], but has at least one pair of complex fixed points, $L$ and $L^*$.

The method is supposed to evaluate superfunction that approaches
$L$ at $\mathrm i \infty$
and approaches
$L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/\mathrm e)$.

and the applicability to other functions required additional investigation.

==Description==

The [[transfer equation]] for the superfunction $F$ of transfer function $T$
Can be written as follows:

$
F(z+1)=T(F(z))
$

Within some domain of values of $z$,
the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z)$

Then, for some strip in direction of imaginary axis,
the transfer equation can be written through the [[integral Cauchi]]

$
F(z)=\oint \frac{F(t)}{t-z)} \mathrm d t
$
where the integration is performed along the boundary of the strip.
In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$;
for long "strip" this is supposed to give the precise approximation of the integral.
The width of the "strip" along the real part of $t$ should be equal to 2;
then, the transfer equation and its inverse allow to represent values
a the edges of the "strip" through the values along the line of the strip.
In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation;
that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.

==References==
<references/>

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. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

Cauchy Method

2016-02-06T13:09:07Z

Kouznetsov:

[[Cauchy Method]] of evaluation of [[superfunction]] refers to the real–holomorphic transfer function $T$ that has no real [[fixed point]], but has at least one pair of complex fixed points, $L$ and $L^*$.

The method is supposed to evaluate superfunction that approaches
$L$ at $\mathrm i \infty$
and
$L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/\mathrm e)$.

and the applicability to other functions required additional investigation.

==Description==

The [[transfer equation]] for the superfunction $F$ of transfer function $T$
Can be written as follows:

$
F(z+1)=T(F(z))
$

Within some domain of values of $z$,
the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z)$

Then, for some strip in direction of imaginary axis,
the transfer equation can be written through the [[integral Cauchi]]

$
F(z)=\oint \frac{F(t)}{t-z)} \mathrm d t
$
where the integration is performed along the boundary of the strip.
In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$;
for long "strip" this is supposed to give the precise approximation of the integral.
The width of the "strip" along the real part of $t$ should be equal to 2;
then, the transfer equation and its inverse allow to represent values
a the edges of the "strip" through the values along the line of the strip.
In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation;
that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.

==References==
<references/>

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. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

Cauchy Method

2016-02-06T13:08:34Z

Kouznetsov: Load stub

[[Cauchy Method]] of evaluation of [[superfunction]] refers to the real–holomorphic transfer function $T$ that has no real [[fixed point]], but has at least one pair of complex fixed points, $L$ and $L*$.

The method is supposed to evaluate superfunction that approaches
$L$ at $\mathrm i \infty$
and
$L^*$ at $-\mathrm i \infty$

Initially, the method had beed designed for $T=\exp_b$ at $b>\exp(1/mathrm e)$.

and the applicability to other functions required additional investigation.

==Description==

The [[transfer equation]] for the superfunction $F$ of transfer function $T$
Can be written as follows:

$
F(z+1)=T(F(z))
$

Within some domain of values of $z$,
the "inverse" equation takes place,

$F(z-1)=T^{-1}(F(z)$

Then, for some strip in direction of imaginary axis,
the transfer equation can be written through the [[integral Cauchi]]

$
F(z)=\oint \frac{F(t)}{t-z)} \mathrm d t
$
where the integration is performed along the boundary of the strip.
In the integrand, value of function $F$ at the top of the strip is replaced to $L$ and that at the bottom is replaced to $L^*$;
for long "strip" this is supposed to give the precise approximation of the integral.
The width of the "strip" along the real part of $t$ should be equal to 2;
then, the transfer equation and its inverse allow to represent values
a the edges of the "strip" through the values along the line of the strip.
In such a way, the suprfunction $F$ can be approximated through the solution of the integral equation;
that can be, in its turn, approximated itertively.

In the simplest case, the contour of integration is just rectangle.

==References==
<references/>

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. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf 
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

==Keywords==
[[Eta]],
[[integral Cauchi]],
[[Tetration]],
[[Transfer equation]]

[[Category:Method Cauchi]]
[[Category:Superfunction]]
[[Category:Tetration]]

File:Penmap.jpg

2015-01-11T05:15:41Z

Kouznetsov: Complex map of natural pentation, $u\!+\!\mathrm i v=\mathrm{pen}(x\!+\! \mathrm i y)$ in the $x,y$ plane. This image is copied from http://mizugadro.mydns.jp/t/index.php/File:Penmap.jpg It is used as figure 7 at the article about [[holomorphic

[[Complex map]] of natural [[pentation]],

$u\!+\!\mathrm i v=\mathrm{pen}(x\!+\! \mathrm i y)$ in the $x,y$ plane.

This image is copied from http://mizugadro.mydns.jp/t/index.php/File:Penmap.jpg
It is used as figure 7 at the article about [[holomorphic ackermanns]]
<ref>

http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf 
http://www.ils.uec.ac.jp/~dima/PAPERS/2014acker.pdf 
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.
</ref>
==References==
<references/>

==[[C++]] generator of curves==

Files [[ado.cin]], [[conto.cin]], [[fsexp.cin]], [[fslog.cin]] should be loaded to the working directory in order to compile the code below
<poem><nomathjax><nowiki>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
#include "fsexp.cin"
#include "fslog.cin"

z_type pen0(z_type z){
DB Lp=-1.8503545290271812;
DB k,a,b;
// k=1.86573322821; a=-.62632418; b=0.4827;
k=1.86573322821; a=-.6263241; b=0.4827;

z_type e=exp(k*z);
return Lp + e*(1.+e*(a+b*e));
}

z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.));
DO(n,7) { if(Re(z)>8.) return 999.; z=FSEXP(z); if(abs(z)<40) goto L1; return 999.; L1: ;}
return z; }

z_type pen(z_type z){ DB x; int m,n;
x=Re(z); if(x<= -4.) return pen0(z);
m=int(x+5.);
z-=DB(m);
z=pen0(z);
DO(n,m) z=FSEXP(z);
return z;
}

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=401,M1=M+1;
int N=801,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
char v[M1*N1]; // v is working array
FILE *o;o=fopen("penma.eps","w"); ado(o,828,828);
fprintf(o,"422 420 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-4.+.02*(m-.5);
DO(n,N1) Y[n]=-4.+.01*(n-.5);
for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,"2 setlinecap .004 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
f[m*N1+n]=9999;}
DO(m,M1){x=X[m];
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=pen0(z);
// c=FSEXP(pen0(z-1.));
// c=FSEXP(FSEXP(pen0(z-2.)));
c=pen7(z);
// d=FSEXP(pen(z-1.));
// p=abs((c-d)/(c+d)); p=-log(p)/log(10.);
p=Re(c); q=Im(c);
if(p>-9999 && p<9999 && fabs(p)>1.e-11) g[m*N1+n]=p;
if(q>-9999 && q<9999 && fabs(q)>1.e-11) f[m*N1+n]=q;
}}
// #include "plofu.cin"

fprintf(o,"1 setlinejoin 2 setlinecap\n");

p=2;q=.5;
for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
fprintf(o,".002 W 0 .6 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
fprintf(o,".002 W .9 0 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
fprintf(o,".002 W 0 0 .9 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".012 W .9 0 0 RGB S\n");
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".012 W 0 0 .9 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".012 W .6 0 .6 RGB S\n");
for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".012 W 0 0 0 RGB S\n");

DB t2=M_PI/1.86573322821;
DB tx=-2.32;

M(tx,t2)L(4.1,t2)
M(tx,-t2)L(4.1,-t2)
fprintf(o,"0 setlinecap .03 W 1 1 1 RGB S\n");
DO(n,64){ x=tx+.1*n; M(x,t2) L(x+.04,t2) }
DO(n,64){ x=tx+.1*n; M(x,-t2) L(x+.04,-t2) }
fprintf(o,"0 setlinecap .04 W 0 0 0 RGB S\n");

//conto(o,g,w,v,X,Y,M,N, ( 1. ),-99,99); fprintf(o,".12 W 1 .5 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);

printf("pen7(-1)=%18.14f\n", Re(pen7(-1.)));
printf("Pi/1.86573322821=%18.14f\n", M_PI/1.86573322821);

system("epstopdf penma.eps");
system( "open penma.pdf");
}
</nowiki></nomathjax></poem>

==[[Latex]] generator of labelw==

<poem><nomathjax><nowiki>
\documentclass[12pt]{article}
\paperheight 832px
\paperwidth 846px
\textwidth 1394px
\textheight 1300px
\topmargin -104px
\oddsidemargin -80px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
{\begin{picture}(824,820)
%\put(12,0){\ing{24}}
\put(12,0){\ing{penma}}
\put(8,808){\sx{3}{$y$}}
\put(8,709){\sx{3}{$3$}}
\put(8,609){\sx{3}{$2$}}
\put(8,509){\sx{3}{$1$}}
\put(8,409){\sx{3}{$0$}}
\put(-12,309){\sx{3}{$-1$}}
\put(-12,209){\sx{3}{$-2$}}
\put(-12,109){\sx{3}{$-3$}}
\put(-12,9){\sx{3}{$-4$}}
\put(4,-8){\sx{3}{$-4$}}
\put(104,-8){\sx{3}{$-3$}}
\put(204,-8){\sx{3}{$-2$}}
\put(304,-8){\sx{3}{$-1$}}
\put(427,-8){\sx{3}{$0$}}
\put(527,-8){\sx{3}{$1$}}
\put(627,-8){\sx{3}{$2$}}
\put(727,-8){\sx{3}{$3$}}
\put(821,-8){\sx{3}{$x$}}
\put(50, 747){\sx{4}{$v\!=\!0$}}
\put(50, 578){\sx{4}{$v\!=\!0$}} \put(760, 580){\sx{4}{\bf cut}}
\put(50, 409){\sx{4}{$v\!=\!0$}}%
\put(50, 240){\sx{4}{$v\!=\!0$}} \put(760, 241){\sx{4}{\bf cut}}
\put(50, 71){\sx{4}{$v\!=\!0$}}
%
\put(326, 638){\sx{4}{$v\!=\!-1$}}
\put(340, 520){\sx{4}{$v\!=\!1$}}
\put(326, 298){\sx{4}{$v\!=\!-1$}}
\put(336, 182){\sx{4}{$v\!=\!1$}}
%
\put(250, 352){\sx{4}{\rot{90}$u\!=\!-1$\ero}}
\put(348, 362){\sx{4}{\rot{90}$u\!=\!0$\ero}}
\put(448, 372){\sx{4}{\rot{90}$u\!=\!1$\ero}}
\put(522, 372){\sx{4}{\rot{90}$u\!=\!2$\ero}}
\end{picture}
\end{document}

</nowiki></nomathjax></poem>

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Talk:Schröder equation

2012-12-15T15:09:41Z

Kouznetsov: Created page with "Henryk, I corrected the misprint. Now I think, how about to define new function $G$, let $$G(z)=\log_c\!\Big(\sigma(z)\Big)$$ ? ~~~~"

Henryk, I corrected the misprint.

Now I think, how about to define new function $G$, let

$$G(z)=\log_c\!\Big(\sigma(z)\Big)$$

? [[User:Kouznetsov|Kouznetsov]] 16:09, 15 December 2012 (CET)

Schröder equation

2012-12-15T15:07:21Z

Kouznetsov:

For a given function $f$ and constant $c$ find a function $\sigma$ such that
$$\sigma(f(x))=c \sigma(x).$$

The Schröder equation can be transformed into the form of an [[Abel equation]]:
$$\log_c\!\Big(\sigma\big(f(x)\big)\Big)=1+\log_c(\sigma(x))$$