imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = [[Complex map]] of function [[Filog]].
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| <math>\mathrm{Filog}(z)</math> expresses the [[fixed point]] of [[logarithm]] to base <math>b\!=\!\exp(z)</math>.
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| Another fixed point to the same base can be expressed with
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| <math>\mathrm{Filog}(z^*)^*</math>
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| ==Algorithm of evaluation==
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| [[Filog]] is expressed through the [[Tania function]]:
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| : <math>\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}</math>
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| ==Representation of the function==
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| <math>f=\mathrm{Filog}(x+\mathrm{i} y)</math> is shown in the <math>x,y</math> plane with
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| levels <math>u=\Re(f)=\mathrm{cont}</math> and
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| levels <math>v=\Im(f)=\mathrm{cont}</math>; thick lines correspond to the integer values.
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| The additional thin gridlines <math>x\!=\!\exp(-1)</math> and <math>x\!=\!\pi/2</math> are drawn. The first of them goes through the branchpoint <math>z=1/\mathrm e</math>, which is the branch point; the second goes through the point
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| <math>z=\pi/2</math>, where the fixed points are </math>\pm \mathrm i</math>.
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| ==Properties of the function==
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| <math>\mathrm{Filog}(z)</math> has two singularities at
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| <math>z\!=\!0</math> and at
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| <math>z\!=\!\exp(-1)</math>; the cutline is directed to the negative part of the real axis.
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| Except the cutline, the function is holomorphic. At the real values of the argument <math>0\!<\!z\!<\!\exp(-1)</math>, both at the upper side of the cut and at the lower side of the cut, the function has real values; in particular, at
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| <math>z=\ln\big(\sqrt{2}\big)</math>, there values are integer
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| <ref name="sqrt2">
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| 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.
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| </ref>:
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| : <math>\mathrm{Filog}(z+\mathrm i o)=2</math>
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| : <math>\mathrm{Filog}(z-\mathrm i o)=4</math>
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| Approaching the branchpoint, the jump at the cut vanishes:
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| : <math> \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</math>
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| ==Generator of curves==
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| // Files [[ado.cin]], [[conto.cin]] and [[filog.cin]] should be loaded to the working directory for the compilation of the [[C++]] code below:
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| #include <math.h>
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| #include <stdio.h>
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| #include <stdlib.h>
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| #define DB double
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| #define DO(x,y) for(x=0;x<y;x++)
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| using namespace std;
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| #include <complex>
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| typedef complex<double> z_type;
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| #define Re(x) x.real()
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| #define Im(x) x.imag()
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| #define I z_type(0.,1.)
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| #include "conto.cin"
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| #include "filog.cin"
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| main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
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| int M=400,M1=M+1;
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| int N=401,N1=N+1;
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| DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
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| char v[M1*N1]; // v is working array
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| FILE *o;o=fopen("filogbig.eps","w");ado(o,2004,2004);
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| fprintf(o,"1002 1002 translate\n 100 100 scale\n");
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| DO(m,M1) X[m]=-10.+.05*(m-.2);
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| DO(n,200)Y[n]=-10.+.05*n;
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| Y[200]=-.0001;
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| Y[201]= .0001;
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| for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
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| for(m=-10;m<11;m++){M(m,-10)L(m,10)}
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| for(n=-10;n<11;n++){M( -10,n)L(10,n)}
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| fprintf(o,".005 W 0 0 0 RGB S\n");
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| M(exp(-1.),-1)
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| L(exp(-1.), 1)
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| M(M_PI/2.,-1)
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| L(M_PI/2., 1)
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| fprintf(o,".003 W 0 0 0 RGB S\n");
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| DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
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| DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
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| DO(n,N1){y=Y[n]; z=z_type(x,y);
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| // c=Tania(z_type(-1.,-M_PI)+log(z))/(-z);
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| c=Filog(z);
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| p=Re(c);q=Im(c);
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| if(p>-15. && p<15. && q>-15. && q<15. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
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| }}
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| fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=3.;q=1;
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| 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");
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| 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");
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| 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");
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| 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");
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| 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");
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| conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".004 W .6 0 .6 RGB S\n");
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| 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");
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| fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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| system("epstopdf filogbig.eps");
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| system( "open filogbig.pdf"); //for mac
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| // getchar(); system("killall Preview"); // for mac
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| // Copyleft 2012 by Dmitrii Kouznetsov
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| }
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| ==Generator of labels==
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| For the compilation of the [[Latex]] source below, the curves of the [[complex map]] should be already generated and stored in file
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| filogbig.pdf with the [[C++]] code above.
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| <nowiki>
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| \documentclass[12pt]{article} %<br>
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| \usepackage{geometry} %<br>
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| \paperwidth 2074pt %<br>
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| \paperheight 2060pt %<br>
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| \topmargin -96pt %<br>
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| \oddsidemargin -80pt %<br>
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| \textwidth 2090pt %<br>
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| \textheight 2066pt %<br>
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| \usepackage{graphicx} %<br>
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| \usepackage{rotating} %<br>
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| \newcommand \rot {\begin{rotate}} %<br>
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| \newcommand \ero {\end{rotate}} %<br>
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| \newcommand \rme {\mathrm{e}} %<br>
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| \newcommand \sx {\scalebox} %<br>
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| \begin{document} %<br>
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| \begin{picture}(2018,2040) %<br>
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| \put(50,40){\includegraphics{filogbig}} %<br>
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| \put(16,2024){\sx{4.3}{</math>y</math>}} %<br>
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| \put(16,1828){\sx{4.2}{</math>8</math>}} %<br>
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| \put(16,1628){\sx{4.2}{</math>6</math>}} %<br>
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| \put(16,1428){\sx{4.2}{</math>4</math>}} %<br>
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| \put(16,1228){\sx{4.2}{</math>2</math>}} %<br>
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| \put(16,1028){\sx{4.2}{</math>0</math>}} %<br>
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| \put(-11,828){\sx{4}{</math>-2</math>}} %<br>
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| \put(-11,628){\sx{4}{</math>-4</math>}} %<br>
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| \put(-11,428){\sx{4}{</math>-6</math>}} %<br>
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| \put(-11,228){\sx{4}{</math>-8</math>}} %<br>
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| \put(-8,0){\sx{4}{</math>-10</math>}} %<br>
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| \put(204,0){\sx{4}{</math>-8</math>}} %<br>
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| \put(404,0){\sx{4}{</math>-6</math>}} %<br>
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| \put(604,0){\sx{4}{</math>-4</math>}} %<br>
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| \put(804,0){\sx{4}{</math>-2</math>}} %<br>
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| \put(1046,0){\sx{4}{</math>0</math>}} %<br>
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| \put(1246,0){\sx{4}{</math>2</math>}} %<br>
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| \put(1446,0){\sx{4}{</math>4</math>}} %<br>
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| \put(1646,0){\sx{4}{</math>6</math>}} %<br>
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| \put(1846,0){\sx{4}{</math>8</math>}} %<br>
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| \put(2036,0){\sx{4.2}{</math>x</math>}} %<br>
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| %\put(40, 2){\sx{.8}{</math>1/\rme</math>}} %<br>
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| %\put(108, 0){\sx{1}{</math>1</math>}} %<br>
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| %\put(164, 2){\sx{.8}{</math>\pi/2</math>}} %<br>
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| \put(1600,1480){\sx{6}{\rot{55}</math>u\!=\!0</math> \ero} } %<br>
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| \put(270,1240){\sx{6}{\rot{60}</math>u\!=\!0.2</math> \ero} } %<br>
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| \put(800,1070){\sx{6}{\rot{55}</math>u\!=\!0.4</math> \ero} } %<br>
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| \put(90,910){\sx{6}{\rot{16}</math>u\!=\!0</math> \ero} } %<br>
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| \put(286,470){\sx{6}{\rot{70}</math>u\!=\!-0.2</math> \ero} } %<br>
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| \put(1686,970){\sx{6}{\rot{-30}</math>u\!=\!-0.2</math> \ero} } %<br>
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| \put(1686,610){\sx{6}{\rot{26}</math>v\!=\!0.2</math> \ero} } %<br>
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| \put(1316,210){\sx{6}{\rot{-56}</math>v\!=\!0</math> \ero} } %<br>
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| \put( 330,444){\sx{6}{\rot{5}</math>v\!=\!-0.4</math> \ero} } %<br>
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| \put( 700,10){\sx{6}{\rot{56}</math>v\!=\!-0.2</math> \ero} } %<br>
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| \end{picture} %<br>
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| \end{document} %<br>
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| %Copyleft 2012 by Dmitrii Kouznetsov
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| </nowiki>
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| The resulting [[PDF]] file is converted to [[PNG]] with 100 pixels/inch resolution.
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| ==Rwfwewnces==
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| <references/>
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| ==Keywords==
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| [[Fixed point]],
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| [[Filog]],
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| [[Tania function]],
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| [[Tetration]],
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| [[Complex map]]
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2012.03.08
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| |pub-country = Japan
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| |notes = I tried to save it as http://en.citizendium.org/wiki/File:FilogmapT.png
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| but it does not load as it is expected..
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| |versions = [[File:FilogmapT.png|30px]] and http://tori.ils.uec.ac.jp/TORI/index.php/File:Filogbigmap100.png
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| }}
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| == Licensing ==
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| {{CC|by|3.0}}
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