Difference between revisions of "File:Loge1emapT1000.jpg"
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| + | {{oq|Loge1emapT1000.jpg|}} |
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| − | Importing image file |
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| + | [[Complex map]] of [[logarithm]] to [[base e1e]], \(b=\exp(1/\mathrm e)\approx 1.444667861\) |
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| + | |||
| + | \[ |
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| + | u+\mathrm i v=\log_{b}(x+\mathrm i y) |
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| + | \] |
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| + | |||
| + | This map appears as Fig.10.4 at page 123 of book |
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| + | «[[Superfunctions]]», 2020 <ref> |
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| + | https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics - 2020/7/28 |
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| + | </ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]]. |
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| + | </ref> |
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| + | in order to explain better which function is treated as [[Transfer function]]. |
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| + | |||
| + | ==[[C++]] generator of curves== |
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| + | //Files [[ado.cin]] and [[conto.cin]] should be loaded to the working directory in order to compile the [[C++]] code below. |
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| + | <pre> |
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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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| + | |||
| + | // DB B=sqrt(2.); |
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| + | DB B=exp(1./M_E); |
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| + | DB LB=1./M_E; |
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| + | |||
| + | int main(){ int j,k,m,n; DB x,y, p,q, t,r; z_type z,c,d; |
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| + | r=log(1./(M_E-1.)); printf("r=%16.14f\n",r); |
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| + | int M=100,M1=M+1; |
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| + | int N=400,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("loge1emap.eps","w");ado(o,162,162); |
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| + | fprintf(o,"81 81 translate\n 10 10 scale\n"); |
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| + | // DO(m,M1) {X[m]=-8.+.04*(m); |
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| + | // DO(m,M1) X[m]=log(exp(-8.)+.02*m*(1.+.3*m)); |
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| + | DO(m,M1) X[m]=4.3* sinh( log(4.)*(-1.+.02*m) ); |
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| + | DO(n,N1) Y[n]=-8.+.04*n; |
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| + | for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}} |
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| + | for(n=-8;n<9;n++){ M( -8,n)L(8,n)} |
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| + | fprintf(o,".008 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=log(z)/LB; |
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| + | // c=exp(LB*z); |
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| + | p=Re(c);q=Im(c); |
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| + | if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;} |
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| + | }} |
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| + | fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=4;q=1.; |
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| + | for(m=-10;m<11;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"); |
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| + | for(m=0;m<11;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"); |
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| + | for(m=0;m<11;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"); |
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| + | for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n"); |
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| + | for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 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,".05 W .6 0 .6 RGB S\n"); |
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| + | for(m=-8;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n"); |
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| + | |||
| + | // for(y=-2*M_PI;y<7.;y+=2*M_PI) |
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| + | y=0.; |
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| + | { |
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| + | M(0,y)L(-8.1,y) fprintf(o,"0 setlinecap .04 W 1 1 1 RGB S\n"); |
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| + | for(m=0;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 1 .5 0 RGB S\n"); |
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| + | for(m=2;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 0 .5 1 RGB S\n"); |
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| + | } |
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| + | |||
| + | fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); |
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| + | system("epstopdf loge1emap.eps"); |
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| + | system( "open loge1emap.pdf"); |
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| + | printf("r=%16.14f %16.14f\n",r,sqrt(M_PI*M_PI+r*r)); |
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| + | getchar(); system("killall Preview"); |
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| + | } |
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| + | </pre> |
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| + | |||
| + | ==[[Latex]] generator of labels== |
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| + | <pre> |
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| + | \documentclass[12pt]{article} |
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| + | \paperwidth 170px |
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| + | \paperheight 168px |
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| + | \textwidth 304px |
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| + | \textheight 300px |
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| + | \topmargin -106px |
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| + | \oddsidemargin -72px |
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| + | \usepackage{graphics} |
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| + | \usepackage{rotating} |
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| + | \newcommand \sx {\scalebox} |
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| + | \newcommand \rot {\begin{rotate}} |
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| + | \newcommand \ero {\end{rotate}} |
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| + | \newcommand \ing {\includegraphics} |
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| + | \newcommand \rmi {\mathrm{i}} |
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| + | \parindent 0pt |
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| + | \pagestyle{empty} |
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| + | \begin{document} |
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| + | \parindent 0pt |
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| + | %\put(40,20){\ing{z2itmap}} |
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| + | %\put(1,1){\ing{expe1emap}} |
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| + | \hskip 6pt |
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| + | \begin{picture}(162,161) % <br> |
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| + | \put(1,1){\ing{loge1emap}} % <br> |
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| + | \put(-3,159.9){\sx{.7}{$y$}} |
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| + | \put(-3,140){\sx{.6}{$6$}} |
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| + | \put(-3,120){\sx{.6}{$4$}} |
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| + | \put(-3,100){\sx{.6}{$2$}} |
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| + | \put(-3,80){\sx{.6}{$0$}} |
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| + | \put(-8,60){\sx{.6}{$-2$}} |
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| + | \put(-8,40){\sx{.6}{$-4$}} |
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| + | \put(-8,20){\sx{.6}{$-6$}} |
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| + | \put(16,-4){\sx{.6}{$-6$}} |
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| + | \put(36,-4){\sx{.6}{$-4$}} |
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| + | \put(56,-4){\sx{.6}{$-2$}} |
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| + | \put(81,-4){\sx{.6}{$0$}} |
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| + | \put(101,-4){\sx{.6}{$2$}} |
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| + | \put(121,-4){\sx{.6}{$4$}} |
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| + | \put(141,-4){\sx{.6}{$6$}} |
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| + | \put(159.6,-4){\sx{.7}{$x$}} |
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| + | \put(090,139.2){\sx{.7}{\rot{82}$v\!=\!4$\ero}} |
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| + | \put(109.6,132){\sx{.7}{\rot{62}$v\!=\!3$\ero}} |
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| + | \put(126,118.6){\sx{.7}{\rot{43}$v\!=\!2$\ero}} |
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| + | \put(136,100.6){\sx{.7}{\rot{21}$v\!=\!1$\ero}} |
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| + | \put(039,138){\sx{.66}{\rot{-54}$v\!=\!6$\ero}} |
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| + | \put(022,118.2){\sx{.66}{\rot{-32}$v\!=\!7$\ero}} |
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| + | \put(013,094){\sx{.66}{\rot{-11}$v\!=\!8$\ero}} |
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| + | \put(140,080.2){\sx{.7}{$v\!=\!0$}} |
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| + | \put(010,066){\sx{.66}{\rot{9}$v\!=\!-8$\ero}} |
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| + | \put(020,039.8){\sx{.66}{\rot{33}$v\!=\!-7$\ero}} |
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| + | \put(039.6,021){\sx{.66}{\rot{53}$v\!=\!-6$\ero}} |
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| + | %\put(140,059){\sx{.7}{\rot{-20}$v\!=\!-1$\ero}} |
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| + | \put(133,060.5){\sx{.66}{\rot{-21}$v\!=\!-1$\ero}} |
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| + | \put(123,042.6){\sx{.66}{\rot{-42}$v\!=\!-2$\ero}} |
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| + | \put(105.4,032){\sx{.66}{\rot{-63}$v\!=\!-3$\ero}} |
||
| + | %\put(091.2,025){\sx{.66}{\rot{-79}$v\!=\!-4$\ero}} |
||
| + | \put(72,066){\sx{.66}{\rot{1}$u\!=\!1$\ero}} |
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| + | \put(72,059){\sx{.66}{\rot{1}$u\!=\!2$\ero}} |
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| + | \put(72,050){\sx{.66}{\rot{1}$u\!=\!3$\ero}} |
||
| + | \put(72,036.6){\sx{.66}{\rot{1}$u\!=\!4$\ero}} |
||
| + | \put(72,017){\sx{.66}{\rot{1}$u\!=\!5$\ero}} |
||
| + | %\put(72,000.2){\sx{.66}{\rot{1}$u\!=\!6$\ero}} |
||
| + | \end{picture} |
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| + | \end{document} |
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| + | </pre> |
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| + | ==References== |
||
| + | {{ref}} |
||
| + | |||
| + | {{fer}} |
||
| + | ==Keywords== |
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| + | |||
| + | «[[]]», |
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| + | «[[Base e1e]]», |
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| + | «[[Exotic iterations]]», |
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| + | «[[Fixed pount]]», |
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| + | «[[Inverse function]]», |
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| + | «[[Logarithm]]», |
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| + | «[[Superfunctions]]», |
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| + | «[[Transfer function]]», |
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| + | «[[Transferfunction]]», |
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| + | |||
| + | [[Category:Base e1e]] |
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| + | [[Category:Book]] |
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| + | [[Category:BookMap]] |
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| + | [[Category:Complex map]] |
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| + | [[Category:Inverse function]] |
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| + | [[Category:Logarithm]] |
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| + | [[Category:Superfuncitons]] |
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| + | [[Category:Transfer function]] |
||
Latest revision as of 16:22, 23 August 2025
Complex map of logarithm to base e1e, \(b=\exp(1/\mathrm e)\approx 1.444667861\)
\[ u+\mathrm i v=\log_{b}(x+\mathrm i y) \]
This map appears as Fig.10.4 at page 123 of book «Superfunctions», 2020 [1][2] in order to explain better which function is treated as Transfer function.
C++ generator of curves
//Files ado.cin and conto.cin should be loaded to the working directory in order to compile 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"
// DB B=sqrt(2.);
DB B=exp(1./M_E);
DB LB=1./M_E;
int main(){ int j,k,m,n; DB x,y, p,q, t,r; z_type z,c,d;
r=log(1./(M_E-1.)); printf("r=%16.14f\n",r);
int M=100,M1=M+1;
int N=400,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("loge1emap.eps","w");ado(o,162,162);
fprintf(o,"81 81 translate\n 10 10 scale\n");
// DO(m,M1) {X[m]=-8.+.04*(m);
// DO(m,M1) X[m]=log(exp(-8.)+.02*m*(1.+.3*m));
DO(m,M1) X[m]=4.3* sinh( log(4.)*(-1.+.02*m) );
DO(n,N1) Y[n]=-8.+.04*n;
for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
for(n=-8;n<9;n++){ M( -8,n)L(8,n)}
fprintf(o,".008 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]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
c=log(z)/LB;
// c=exp(LB*z);
p=Re(c);q=Im(c);
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
}}
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=4;q=1.;
for(m=-10;m<11;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<11;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<11;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<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n");
for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 .9 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".05 W .6 0 .6 RGB S\n");
for(m=-8;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
// for(y=-2*M_PI;y<7.;y+=2*M_PI)
y=0.;
{
M(0,y)L(-8.1,y) fprintf(o,"0 setlinecap .04 W 1 1 1 RGB S\n");
for(m=0;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 1 .5 0 RGB S\n");
for(m=2;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 0 .5 1 RGB S\n");
}
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf loge1emap.eps");
system( "open loge1emap.pdf");
printf("r=%16.14f %16.14f\n",r,sqrt(M_PI*M_PI+r*r));
getchar(); system("killall Preview");
}
Latex generator of labels
\documentclass[12pt]{article}
\paperwidth 170px
\paperheight 168px
\textwidth 304px
\textheight 300px
\topmargin -106px
\oddsidemargin -72px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\parindent 0pt
\pagestyle{empty}
\begin{document}
\parindent 0pt
%\put(40,20){\ing{z2itmap}}
%\put(1,1){\ing{expe1emap}}
\hskip 6pt
\begin{picture}(162,161) % <br>
\put(1,1){\ing{loge1emap}} % <br>
\put(-3,159.9){\sx{.7}{$y$}}
\put(-3,140){\sx{.6}{$6$}}
\put(-3,120){\sx{.6}{$4$}}
\put(-3,100){\sx{.6}{$2$}}
\put(-3,80){\sx{.6}{$0$}}
\put(-8,60){\sx{.6}{$-2$}}
\put(-8,40){\sx{.6}{$-4$}}
\put(-8,20){\sx{.6}{$-6$}}
\put(16,-4){\sx{.6}{$-6$}}
\put(36,-4){\sx{.6}{$-4$}}
\put(56,-4){\sx{.6}{$-2$}}
\put(81,-4){\sx{.6}{$0$}}
\put(101,-4){\sx{.6}{$2$}}
\put(121,-4){\sx{.6}{$4$}}
\put(141,-4){\sx{.6}{$6$}}
\put(159.6,-4){\sx{.7}{$x$}}
\put(090,139.2){\sx{.7}{\rot{82}$v\!=\!4$\ero}}
\put(109.6,132){\sx{.7}{\rot{62}$v\!=\!3$\ero}}
\put(126,118.6){\sx{.7}{\rot{43}$v\!=\!2$\ero}}
\put(136,100.6){\sx{.7}{\rot{21}$v\!=\!1$\ero}}
\put(039,138){\sx{.66}{\rot{-54}$v\!=\!6$\ero}}
\put(022,118.2){\sx{.66}{\rot{-32}$v\!=\!7$\ero}}
\put(013,094){\sx{.66}{\rot{-11}$v\!=\!8$\ero}}
\put(140,080.2){\sx{.7}{$v\!=\!0$}}
\put(010,066){\sx{.66}{\rot{9}$v\!=\!-8$\ero}}
\put(020,039.8){\sx{.66}{\rot{33}$v\!=\!-7$\ero}}
\put(039.6,021){\sx{.66}{\rot{53}$v\!=\!-6$\ero}}
%\put(140,059){\sx{.7}{\rot{-20}$v\!=\!-1$\ero}}
\put(133,060.5){\sx{.66}{\rot{-21}$v\!=\!-1$\ero}}
\put(123,042.6){\sx{.66}{\rot{-42}$v\!=\!-2$\ero}}
\put(105.4,032){\sx{.66}{\rot{-63}$v\!=\!-3$\ero}}
%\put(091.2,025){\sx{.66}{\rot{-79}$v\!=\!-4$\ero}}
\put(72,066){\sx{.66}{\rot{1}$u\!=\!1$\ero}}
\put(72,059){\sx{.66}{\rot{1}$u\!=\!2$\ero}}
\put(72,050){\sx{.66}{\rot{1}$u\!=\!3$\ero}}
\put(72,036.6){\sx{.66}{\rot{1}$u\!=\!4$\ero}}
\put(72,017){\sx{.66}{\rot{1}$u\!=\!5$\ero}}
%\put(72,000.2){\sx{.66}{\rot{1}$u\!=\!6$\ero}}
\end{picture}
\end{document}
References
- ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
- ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.
Keywords
«[[]]», «Base e1e», «Exotic iterations», «Fixed pount», «Inverse function», «Logarithm», «Superfunctions», «Transfer function», «Transferfunction»,
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 06:13, 1 December 2018 |  | 2,361 × 2,333 (1.67 MB) | Maintenance script (talk | contribs) | Importing image file |
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