File:Sqrt2figf45bT.png

Complex map of the primary expansion of the growing superexponential to base $b\!=\!\sqrt{2}$ built up at the fixed point $L\!=\!4$.

For this plot, 20 terms are taken into account in the series. The argument is displaced for $x_{45}=-1.11520724513161$ in order to simplify the comparison with the superfunction $F$, constructed with the regular iteration, such that $F(0)=5$.


 * $ u\!+\! \mathrm i v = \tilde f\Big( x\!-\!1.11520724513161 +\mathrm i \, y \Big)$


 * $ \displaystyle \tilde f(z)= \sum_{n=0}^{N-1} a_n \exp(knz)$

for $N=20$.


 * $a_0=L=4$ is fixed point of the exponential transfer function $T$,

$T(z)=\exp_{\sqrt{2}}(z)=\Big( \sqrt{2} \Big)^z$


 * $ a_1=1$

Value of $k$ and other coefficients are found substituting the expansion into the Transfer equation $F(z\!+\!1)=T\Big(F(z)\Big)$; the approximations can be extracted also from article or from the code in the next section.

The image is almost figure 4a of publication in Mathematics of Computation ; the notations are a little bit simplified.

C++ generator of curves
using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) include "conto.cin"

z_type f45E(z_type z){int n; z_type e,s; DB coeu[21]={1., 0.44858743119526122890, .19037224679780675668, 0.77829576536968278770e-1, 0.30935860305707997953e-1, 0.12022125769065893274e-1, 0.45849888965617461424e-2, 0.17207423310577291102e-2, 0.63681090387985537364e-3, 0.23276960030302567773e-3, 0.84145511838119915857e-4, 0.30115646493706434120e-4, 0.10680745813035087964e-4, 0.37565713615564248453e-5, 0.13111367785052622794e-5, 0.45437916254218158081e-6, 0.15642984632975371803e-6, 0.53523276400816416929e-7, 0.18207786280204973113e-7, 0.61604764947389226744e-8, 0.2e-8}; e=exp(.32663425997828098238*(z-1.11520724513161)); //     s=coeu[20]; for(n=19;n>=0;n--) { s*=e; s+=coeu[n]; } s=coeu[19]; for(n=18;n>=0;n--) { s*=e; s+=coeu[n]; } return 4.+s*e;} z_type F45E(z_type z){ DB b=sqrt(2.); if(Re(z)<-1.) return f45E(z); return exp(F45E(z-1.)*log(b)); }

// #include"sqrt2f45E.cin" main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=501,M1=M+1; int N=501,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("sqrt2figf45b.eps","w"); ado(o,202,202); fprintf(o,"101 101 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-10.+.033*(m-.5); DO(n,N1) Y[n]=-10.+.04*(n-.5); for(m=-10;m<11;m++) {M(m,-10)L(m,10)} for(n=-10;n<11;n++) {M( -10,n)L(10,n)} fprintf(o,"1 setlinejoin 2 setlinecap\n"); fprintf(o," .006 W 0 0 0 RGB S\n"); // z_type tm,tp,F[M1*N1]; 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=f45E(z); p=Re(c); q=Im(c); if(p>-25. && p<25. && q>-25. && q<25.) { g[m*N1+n]=p; f[m*N1+n]=q;} }} p=2.5; q=.5; for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".01 W 0 .6 0 RGB S\n"); for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q); fprintf(o,".01 W .9 0 0 RGB S\n"); for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".01 W 0 0 .9 RGB S\n"); for(m= 1;m<16;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n"); for(m= 1;m<16;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n"); for(m=-31;m<16;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n"); // #include "plofu.cin" // p=1.e-15; //for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.16f\n",q, Re(uq2e(z))  ); } //M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf sqrt2figf45b.eps"); system(   "open sqrt2figf45b.pdf"); //for macintosh }

Latex generator of labels
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