File:FilogbigT.jpg

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Summary

Title / Description	Complex map of function Filog. ==Semantics of Filog== $\mathrm{Filog}(z)$ expresses the fixed point of logarithm to base $b\!=\!\exp(z)$. Another fixed point to the same base can be expressed with $\mathrm{Filog}(z^)^$ ==Algorithm of evaluation== Filog is expressed through the Tania function: : $\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}$ ==Representation of the function== $f=\mathrm{Filog}(x+\mathrm{i} y)$ is shown in the $x,y$ plane with levels $u=\Re(f)=\mathrm{cont}$ and levels $v=\Im(f)=\mathrm{cont}$; thick lines correspond to the integer values. The additional thin gridlines $x\!=\!\exp(-1)$ and $x\!=\!\pi/2$ are drawn. The first of them goes through the branchpoint $z=1/\mathrm e$, which is the branch point; the second goes through the point $z=\pi/2$, where the fixed points are $\pm \mathrm i$. ==Properties of the function== $\mathrm{Filog}(z)$ has two singularities at $z\!=\!0$ and at $z\!=\!\exp(-1)$; the cutline is directed to the negative part of the real axis. Except the cutline, the function is holomorphic. At the real values of the argument $0\!<\!z\!<\!\exp(-1)$, both at the upper side of the cut and at the lower side of the cut, the function has real values; in particular, at $z=\ln\big(\sqrt{2}\big)$, there values are integer ^[1]: : $\mathrm{Filog}(z+\mathrm i o)=2$ : $\mathrm{Filog}(z-\mathrm i o)=4$ Approaching the branchpoint, the jump at the cut vanishes: : $ \displaystyle \lim_{x\rightarrow 1/\mathrm e} \mathrm{Filog}(x+\mathrm i o)= \lim_{x\rightarrow 1/\mathrm e} \mathrm{Filog}(x-\mathrm i o)= \mathrm e$ ==Generator of curves== // Files ado.cin, conto.cin and filog.cin should be loaded to the working directory for the compilation of the C++ code below: #include <math.h> #include <stdio.h> #include <stdlib.h> #define DB double #define DO(x,y) for(x=0;x<y;x++) using namespace std; #include <complex> typedef complex<double> z_type; #define Re(x) x.real() #define Im(x) x.imag() #define I z_type(0.,1.) #include "conto.cin" #include "filog.cin" main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=400,M1=M+1; int N=401,N1=N+1; DB X[M1],Y[N1], g[M1N1],f[M1N1], w[M1N1]; // w is working array. char v[M1N1]; // v is working array FILE o;o=fopen("filogbig.eps","w");ado(o,2004,2004); fprintf(o,"1002 1002 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-10.+.05(m-.2); DO(n,200)Y[n]=-10.+.05n; Y[200]=-.0001; Y[201]= .0001; for(n=202;n<N1;n++) Y[n]=-10.+.05(n-1.); 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,".005 W 0 0 0 RGB S\n"); M(exp(-1.),-1) L(exp(-1.), 1) M(M_PI/2.,-1) L(M_PI/2., 1) fprintf(o,".003 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[mN1+n]=9999; f[mN1+n]=9999;} DO(m,M1){x=X[m]; //printf("%5.2f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); // c=Tania(z_type(-1.,-M_PI)+log(z))/(-z); c=Filog(z); p=Re(c);q=Im(c); if(p>-15. && p<15. && q>-15. && q<15. ){ g[mN1+n]=p;f[mN1+n]=q;}
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fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=3.;q=1; for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".001 W 0 .6 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".001 W .9 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".001 W 0 0 .9 RGB S\n"); for(m=1;m<14;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".004 W .9 0 0 RGB S\n"); for(m=1;m<14;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".004 W .6 0 .6 RGB S\n"); for(m=-11;m<14;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf filogbig.eps"); system( "open filogbig.pdf"); //for mac // getchar(); system("killall Preview"); // for mac // Copyleft 2012 by Dmitrii Kouznetsov } ==Generator of labels== For the compilation of the Latex source below, the curves of the complex map should be already generated and stored in file fIlog.pdf with the C++ code above. \documentclass[12pt]{article} % \usepackage{geometry} % \paperwidth 2074pt % \paperheight 2060pt % \topmargin -96pt % \oddsidemargin -80pt % \textwidth 2090pt % \textheight 2066pt % \usepackage{graphicx} % \usepackage{rotating} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \rme {\mathrm{e}} % \newcommand \sx {\scalebox} % \begin{document} % \begin{picture}(2018,2040) % \put(50,40){\includegraphics{filogbig}} % \put(16,2024){\sx{4.3}{$y$}} % \put(16,1828){\sx{4.2}{$8$}} % \put(16,1628){\sx{4.2}{$6$}} % \put(16,1428){\sx{4.2}{$4$}} % \put(16,1228){\sx{4.2}{$2$}} % \put(16,1028){\sx{4.2}{$0$}} % \put(-11,828){\sx{4}{$-2$}} % \put(-11,628){\sx{4}{$-4$}} % \put(-11,428){\sx{4}{$-6$}} % \put(-11,228){\sx{4}{$-8$}} % \put(-8,0){\sx{4}{$-10$}} % \put(204,0){\sx{4}{$-8$}} % \put(404,0){\sx{4}{$-6$}} % \put(604,0){\sx{4}{$-4$}} % \put(804,0){\sx{4}{$-2$}} % \put(1046,0){\sx{4}{$0$}} % \put(1246,0){\sx{4}{$2$}} % \put(1446,0){\sx{4}{$4$}} % \put(1646,0){\sx{4}{$6$}} % \put(1846,0){\sx{4}{$8$}} % \put(2036,0){\sx{4.2}{$x$}} % %\put(40, 2){\sx{.8}{$1/\rme$}} % %\put(108, 0){\sx{1}{$1$}} % %\put(164, 2){\sx{.8}{$\pi/2$}} % \put(1600,1480){\sx{6}{\rot{55}$u\!=\!0$ \ero} } % \put(270,1240){\sx{6}{\rot{60}$u\!=\!0.2$ \ero} } % \put(800,1070){\sx{6}{\rot{55}$u\!=\!0.4$ \ero} } % \put(90,910){\sx{6}{\rot{16}$u\!=\!0$ \ero} } % \put(286,470){\sx{6}{\rot{70}$u\!=\!-0.2$ \ero} } % \put(1686,970){\sx{6}{\rot{-30}$u\!=\!-0.2$ \ero} } % \put(1686,610){\sx{6}{\rot{26}$v\!=\!0.2$ \ero} } % \put(1316,210){\sx{6}{\rot{-56}$v\!=\!0$ \ero} } % \put( 330,444){\sx{6}{\rot{5}$v\!=\!-0.4$ \ero} } % \put( 700,10){\sx{6}{\rot{56}$v\!=\!-0.2$ \ero} } % \end{picture} % \end{document} % %Copyleft 2012 by Dmitrii Kouznetsov The resulting PDF file is converted to PNG with 100 pixels/inch resolution. ==Rwfwewnces==

↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

|author = Dmitrii Kouznetsov |date-created = 2012.03.08 |pub-country = Japan |notes = I tried to save it as http://en.citizendium.org/wiki/File:FilogmapT.png but it does not load as it is expected.. |versions = File:FilogmapT.png and http://tori.ils.uec.ac.jp/TORI/index.php/File:Filogbigmap100.png }}

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Fixed point of logarithm

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[sqrt2-1] ttp://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

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