File:TetSheldonImaT.png

Few iterations at the Iterated Cauchi for the Tetration to Sheldon base.

The holomorphic solution $F$ of equation
 * $ \!\!\!\!\!\!\!\! (1) ~ ~ ~

\exp(a F(z))=F(z\!+\! 1)$ is considered for
 * $\!\!\!\!\!\! (2) ~ ~ ~

a=\ln(s) = \ln( 1.52598338517+0.0178411853321 i) \approx 0.4227073870410604+0.0116910660021443 i $ where $s$ is the Sheldon number.

The solution $F$ is supposed to have the specific boundary behavior:
 * $\!\!\!\!\!\! (3) ~ ~ ~

F(x+ i \infty) = Z_{\mathrm o}=\mathrm{Filog}(a)$
 * $\!\!\!\!\!\! (4) ~ ~ ~

F(x- i \infty) = Z_{\mathrm c}=\mathrm{Filog}(a^*)^*$ where Filog is fixed point of the logarithm.

Approximation
The solution is approximated first along the imaginary axis,
 * $\!\!\!\!\!\! (5) ~ ~ ~ F(iy)=f(x)$

where $f$ is solution of equation
 * $\!\!\!\!\!\! (5) ~ ~ ~ \displaystyle

f(x) = K(ix) + \frac{1}{2\pi} \int_{-A}^{A} \frac{ \exp(a f(y) ) }{ 1+ \mathrm i y- \mathrm i x} \mathrm d y + \frac{1}{2\pi} \int_{-A}^{A} \frac{ \log(f(y))/a }{-1+\mathrm i y- \mathrm i x} \mathrm d y $ with
 * $\!\!\!\!\!\! (6) ~ ~ ~ \displaystyle

K(z)= Z_{\mathrm o} \left(\frac 12-\frac{1}{2\pi\mathrm i} \ln \frac {1-\mathrm i A +z} {1+\mathrm i A-z} \right) +  \displaystyle Z_{\mathrm c} \left( \frac 12-\frac{1}{2\pi\mathrm i} \ln  \frac {1-\mathrm i A -z} {1+\mathrm i A+z} \right) $

where $A$ is large positive constant.

The Gauss-Legendre quadrature formula is used for the numerical implementation of the integration with number of nodes 2048. The update by (5) is realized node by node; first even nodes from left to right; then odd ones from right to left. The details can be revealed from the algorithm below.

Generators of the figure
The follpwing files should be loaded to the working directory for the compilation of the generator below: ado.cin (header for graphics) filog.cin (evaluation of Filog) GLxw2048.inc (nodes and weights of the Gauss-Kegendre quadrature formula

C++ generator of curves
using namespace std; typedef complex z_type; main{ int J,j,k,m,n; DB x,y, u, t; z_type z,c,d, cu,cd; z_type b=z_type( 1.5259833851700000, 0.0178411853321000); z_type a=log(b); z_type Zo=Filog(a); z_type Zc=conj(Filog(conj(a))); int K=NPO; DB A=32.; printf("K=%3d A=%3.1f\n",K,A); z_type E[K],F[K],G[K],H[K]; FILE *o;o=fopen("TetSheldonIma.eps","w");ado(o,2200,450); fprintf(o,"1100 201 translate\n 100 100 scale\n"); for(j=-10;j<11;j+=1){M(j,-2)L(j,2);} M(-10, 2)L(10 , 2); M(-10, 1)L(10 , 1); M(-10 ,-1)L(10 ,-1); M(-10 ,-2)L(10 ,-2); fprintf(o,".006 W S\n"); M(-10.1,0)L(10.1,0); fprintf(o,".02 W S\n"); fprintf(o,".01 W S\n 1 setlinejoin 1\n"); DO(n,K){y=GLx[n]*A; if(y<-2.4) 	E[n]=F[n]=G[n]=Zc; else {if(y>2.4)	E[n]=F[n]=G[n]=Zo; else { F[n]=c=1. ;//f3(z_type(0.,y)); E[n]=log(c)/a; G[n]=exp(a*c); } 	    } 	} DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Re(F[k]);if(j==0)M(y,u)else L(y,u)} fprintf(o,".08 W 1 .2 1 RGB S\n"); DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Im(F[k]);if(j==0)M(y,u)else L(y,u)} fprintf(o,".08 W 1 .6 1 RGB S\n"); for(n=0;n<K;n+=2) { y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c)/a; F[n]=c; G[n]=exp(c*a);} DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Re(F[k]);if(j==0)M(y,u)else L(y,u)} DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Im(F[k]);if(j==0)M(y,u)else L(y,u)} fprintf(o,".01 W 1 0 0 RGB S\n"); for(n=K-1;n>0;n-=2) {y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c)/a; F[n]=c;  G[n]=exp(c*a); } DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Re(F[k]);if(j==0)M(y,u)else L(y,u)} DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Im(F[k]);if(j==0)M(y,u)else L(y,u)} fprintf(o,".02 W 0 1 0 RGB S\n"); for(n=0;n<K;n+=2) {y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c)/a; F[n]=c; G[n]=exp(c*a);	} DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Re(F[k]);if(j==0)M(y,u)else L(y,u)} DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Im(F[k]);if(j==0)M(y,u)else L(y,u)} fprintf(o,".01 W 0 0 1 RGB S\n"); for(n=K-1;n>0;n-=2) {y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<4)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c)/a; F[n]=c; G[n]=exp(c*a);} DO(j,4) { for(n=0;n0;n-=2) { y=GLx[n]*A; z=z_type(0.,y); c=0.; //printf(" %3d",n); DO(k,K){t=A*GLx[k]; c+= GLw[k]*( G[k]/(z_type( 1.,t)-z) - E[k]/(z_type(-1.,t)-z) );} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; if(abs(n-K/2)<2)printf("%3d %6.3f %9.6f %9.6f  %9.6f %9.6f\n",n,y,Re(F[n]),Im(F[n]),Re(c),Im(c)); E[n]=log(c)/a; F[n]=c; G[n]=exp(c*a);} } DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Re(F[k]);if(j==0)M(y,u)else L(y,u)} DO(j,440){k=K/2+j-220; y=GLx[k]*A; u=Im(F[k]);if(j==0)M(y,u)else L(y,u)} fprintf(o,".01 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n\%c%cTrailer",'%','%'); fclose(o); system("epstopdf TetSheldonIma.eps"); system(   "open TetSheldonIma.pdf"); o=fopen("TetSheldonIma.inc","w"); fprintf(o,"z_type F[K]={\n"); DO(k,K-1) fprintf(o,"z_type(%19.16lf,%19.16lf),\n",Re(F[k]),Im(F[k])); fprintf(o,"z_type(%19.16lf,%19.16lf)};\n",Re(F[K-1]),Im(F[K-1])); fclose(o); }
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 1) include
 * 1) define Re(x) (x).real
 * 2) define Im(x) (x).imag
 * 3) define I z_type(0.,1.)
 * 4) define DO(x,y) for(x=0;x<y;x++)
 * 5) include "ado.cin"
 * 6) include "filog.cin"
 * 7) define M(x,y) fprintf(o,"%5.3f %5.3f M\n",1.*(x),1.*(y));
 * 8) define L(x,y) fprintf(o,"%5.3f %5.3f L\n",1.*(x),1.*(y));
 * 9) define o(x,y) fprintf(o,"%5.3f %5.3f o\n",1.*(x),1.*(y));
 * 1) include "GLxw2048.inc"

Latex generator of lables
\documentclass[12pt]{article} % \paperwidth 2100pt % \paperheight 472pt % \textwidth 2060pt % \textheight 500pt % \topmargin -100pt % \oddsidemargin -104pt % \pagestyle{empty} % \usepackage[usenames]{color} % \usepackage[utf8x]{inputenc} % \usepackage{hyperref} % \usepackage{graphicx} % \usepackage{rotating} % \newcommand \sx {\scalebox} % \newcommand \ing {\includegraphics} % \newcommand \rme {\mathrm{e}} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \begin{document} % \parindent 0pt % \begin{picture}(2090,432) % \put(-10,10){\ing{TetSheldonIma}} % \put(50,392){\sx{5}{$2$}} % \put(50,293){\sx{5}{$1$}} % \put(50,194){\sx{5}{$0$}} % \put(23,94){\sx{5}{$-\!1$}} % \put(23,-6){\sx{5}{$-\!2$}} % \put(250,-28){\sx{4.5}{$-8$}} % \put(450,-28){\sx{4.5}{$-6$}} % \put(650,-28){\sx{4.5}{$-4$}} % \put(850,-28){\sx{4.5}{$-2$}} % \put(1084,-28){\sx{4.5}{$0$}} % \put(1284,-28){\sx{4.5}{$2$}} % \put(1484,-28){\sx{4.5}{$4$}} % \put(1684,-28){\sx{4.5}{$6$}} % \put(1884,-28){\sx{4.5}{$8$}} % \put(2084,-28){\sx{4.5}{$x$}} % \put(220,394){\sx{4.2}{$\Im\big(f(x)\big)$}} % \put(220,32){\sx{4.2}{$\Re\big(f(x)\big)$}} % \put(1800,377){\sx{4.2}{$\Im\big(f(x)\big)$}} % \put(1800,290){\sx{4.2}{$\Re\big(f(x)\big)$}} % \end{picture} % \end{document}

Related pictures
The same algorithm is used for the evaluation of tetration to the natural base ($b\!=\!\mathrm e$) .

Keywords
Tetration, Tetration to Sheldon base. Filog, Superfunction