Difference between revisions of "File:Expitemap.jpg"

Latest revision as of 20:56, 14 December 2025

Fig.15.5 at page 214 of book «Superfunctions»^[1], 2020:

Complex maps of iterates of natural exponent (exponential); \(\ u\!+\!\mathrm i v=\exp^n(x\!+\!\mathrm i y)\)

The maps appear also in the Russian version «Суперфункции» (2014) ^[2], Рис.15.5, стр. 213.

The two maps at the top refers to \(n\!=\!1\) and \(n\!=\!-1\). They correspond to \(\exp^1\!=\!\exp\) and \(\exp^{-1}\!=\!\ln\).

For \(n=0.5\), the detailed map of the Square root of exponential is loaded as https://mizugadro.mydns.jp/t/index.php/File:QexpMap1263x1243.png

Case \(n=0\) (identity function) is not presented as obvious.

Routines to evaluate the non-integer iterates of exponential are described also in the Vladikavkaz Mathematical Journal ^[3].

The iterates of exponential are expressed through the natural tetration \(\mathrm{tet}\) and arctetration \(\mathrm{tet}^{-1}\!=\!\mathrm{ate}\): \[ \exp^n(z)=\mathrm{tet}\big(n+\mathrm{ate}(z)\big) \] In this expression, number \(n\) of the iterate has no need to be integer.

The C++ implementation of functions tet and ate are loaded as fsexp.cin and fslog.cin.
The titles of these articles may look not so good. Names tet.cin and ate.cin (not loaded yet) are reserved for the optimized versions of these routines. The performance of the primary approximations can be significantly improved. This improvement can be achieved by the cost of reduction of region of the overlapping of the ranges of their validities. This overlapping is necessary for the numerical testing of the primary approximations of these functions.

C++ generator of the First map

/* As the codes to generate the maps are very similar, I load the only generator of the first map.
Files ado.cin, conto.cin, fsexp.cin, fslog.cin should be loaded in order to compile the 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 std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
 #include "fsexp.cin"
 #include "fslog.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=401,M1=M+1;
 int N=201,N1=N+1;
 DB X[M1],Y[N1];
 DB *g, *f, *w; // w is working array.
 g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("exp10map.eps","w");  ado(o,802,402);
 fprintf(o,"401 1 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 DO(m,M1) X[m]=-4.+.02*(m-.5);
 DO(n,N1) Y[n]=0.+.02*(n-.5); 
// for(n=0;n<N1;n++) Y[n]=sinh((2./200.)*(n-200.5));
 for(m=-4;m<5;m++) {M(m,0)L(m,4)}
 for(n=0;n<5;n++) {M(  -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 //fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
 DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                        f[m*N1+n]=999;}
 DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
       { //     c=exp(z);
        c=FSEXP(1.+FSLOG(z));
         p=Re(c); q=Im(c);
        // if(p>-12 && p<12 && fabs(q)>1.e-12) 
        g[m*N1+n]=p;
        // if(q>-12 && q<12 && fabs(q)>1.e-12) 
        f[m*N1+n]=q;
       }
        }}
 fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=2.;q=1;
 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,".007 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,".007 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,".007 W 0 0 .9 RGB S\n");
 for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
 for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
 for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");

conto(o,f,w,v,X,Y,M,N, 1.3372357014306895, -p,p);fprintf(o,".005 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, .31813150520476413, -p,p);fprintf(o,".005 W 0 0 0 RGB S\n");

 fprintf(o,"0 setlinejoin 0 setlinecap\n");
 fprintf(o,"showpage\n");
 fprintf(o,"%c%cTrailer\n",'%','%');
 fclose(o);
 free(f);
 free(g);
 free(w);
       system("epstopdf exp10map.eps"); 
       system(    "open exp10map.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }

Latex combiner

\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 438px
%\paperheight 134px 
\paperheight 682px 
\topmargin -107pt
\oddsidemargin -84pt
\textheight 800px
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\newcommand \LeLa {
\put(-24,384){\sx{4}{$y$}}
\put(-24,288){\sx{4}{$3$}}
\put(-24,188){\sx{4}{$2$}}
\put(-24,88){\sx{4}{$1$}}
\put(-24,-12){\sx{4}{$0$}}
}
\newcommand \DoLa {
\put(70,-36){\sx{3.8}{$-3$}}
\put(170,-36){\sx{3.8}{$-2$}}
\put(270,-36){\sx{3.8}{$-1$}}
\put(390,-36){\sx{3.8}{$0$}}
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\put(590,-36){\sx{3.8}{$2$}}
\put(690,-36){\sx{3.8}{$3$}}
\put(786,-36){\sx{3.8}{$x$}}
}
%\begin{figure}%[H]
~
%\sx{.294}{\begin{picture}(802,402)
\sx{.26}{\begin{picture}(802,402) 
\put(0,0){\ing{exp10map}} \LeLa \put(20,342){\sx{7}{$n\!=\!1$}} 
\put(46,300){\rot{0}\sx{5}{$v\!=\!0$} \ero}
\put(50,144){\rot{0}\sx{5}{$u\!=\!0$} \ero}
\put(46,-13){\rot{0}\sx{5}{$v\!=\!0$} \ero}
\put(260,220){\rot{90}\sx{4.5}{$u\!=\!-0.2$} \ero}
\put(424,224){\rot{90}\sx{5}{$u\!=\!-1$} \ero} 
\put(244,4){\rot{56}\sx{4.5}{$u\!=\!0.2$} \ero} 
\put(404,4){\rot{56}\sx{5}{$u\!=\!1$} \ero}
\put(474,4){\rot{56}\sx{5}{$u\!=\!2$} \ero} 
\end{picture}} ~ 
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm10map}} \put(20,340){\sx{7}{$n\!=\!-1$}} 

\put(30,144){\rot{56} \sx{5}{$u\!=\!1.4$}\ero} %%%%%%
\put(100,114){\rot{55} \sx{5}{$u\!=\!1.2$}\ero} %%%%%%
\put(160,102){\rot{54} \sx{5}{$u\!=\!1$}\ero} 

\put(2,42){\rot{-8} \sx{5}{$v\!=\!3$}\ero}
\put(546,196){\rot{56} \sx{5}{$v\!=\!1$}\ero}
\put(580,162){\rot{45} \sx{5}{$v\!=\!0.8$}\ero} 
\put(610,122){\rot{34} \sx{5}{$v\!=\!0.6$}\ero} 
\put(626,79){\rot{22} \sx{5}{$v\!=\!0.4$}\ero} 
\put(640,34){\rot{11} \sx{5}{$v\!=\!0.2$}\ero} 
\put(641,-12){\rot{0} \sx{5}{$v\!=\!0$}\ero} 
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp09map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.9$}}
\put(50,202){\rot{20}\sx{5}{$v\!=\!0$} \ero}
\put(400,182){\rot{56}\sx{5}{$v\!=\!1$} \ero} 
\put(10,122){\sx{5}{\bf cut}} 
\put(178,2){\rot{54}\sx{5}{$u\!=\!0$} \ero}
\put(430,146){\rot{7}\sx{5}{$u\!=\!0$} \ero}
\put(432, 46){\rot{43}\sx{5}{$u\!=\!1$} \ero}
\put(482, 10){\rot{52}\sx{5}{$u\!=\!2$} \ero}
 \end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm09map}}  \put(20,340){\sx{7}{$n\!=\!-0.9$}} 
\put(180,198){\rot{27} \sx{5}{$u\!=\!1$} \ero}
\put(270,14){\rot{47} \sx{5}{$u\!=\!0$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(606,250){\rot{48} \sx{5}{$v\!=\!1$} \ero} 
\put(636,134){\rot{29} \sx{5}{$v\!=\!0.6$} \ero}
\put(644,84){\rot{19} \sx{5}{$v\!=\!0.4$} \ero}
\put(648,32){\rot{10} \sx{5}{$v\!=\!0.2$} \ero}
\put(649,-13){\rot{0} \sx{5}{$v\!=\!0$}\ero} 
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp08map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.8$}}
\put(168,198){\rot{35}\sx{5}{$v\!=\!0$} \ero}
\put(390,182){\rot{66}\sx{5}{$v\!=\!1$} \ero} 
\put(430,146){\rot{13}\sx{5}{$u\!=\!0$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(40,0){\rot{46}\sx{3.8}{$u\!=\!-0.2$} \ero}
\put(242,2){\rot{54}\sx{5}{$u\!=\!0$} \ero}
\put(432, 42){\rot{46}\sx{5}{$u\!=\!1$} \ero}
\put(488, 1){\rot{59}\sx{5}{$u\!=\!2$} \ero}
 \end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm08map}}  \put(20,340){\sx{7}{$n\!=\!-0.8$}} 
\put(184,192){\rot{27} \sx{5}{$u\!=\!1$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(230,36){\rot{34} \sx{5}{$u\!=\!0$} \ero}
\put(6,1){\rot{52} \sx{4}{$u\!=\!0.4$} \ero}
\put(166,3){\rot{54} \sx{4}{$u\!=\!0.2$} \ero}
\put(506,234){\rot{53} \sx{4}{$v\!\approx\!1.337$} \ero} 
\put(606,220){\rot{42} \sx{5}{$v\!=\!1$} \ero} 
\put(636,120){\rot{26} \sx{5}{$v\!=\!0.6$} \ero}
\put(644,76){\rot{16} \sx{5}{$v\!=\!0.4$} \ero}
\put(648,30){\rot{8} \sx{5}{$v\!=\!0.2$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp05map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.5$}} 
\put(360,294){\rot{11} \sx{5}{$u\!=\!-2$} \ero} 
\put(382,222){\rot{22} \sx{5}{$u\!=\!-1$} \ero} 
\put(420,146){\rot{38} \sx{5}{$u\!=\!0$} \ero} 
\put(490,99){\rot{54} \sx{5}{$u\!=\!1$} \ero} 
\put(550,82){\rot{58} \sx{5}{$u\!=\!2$} \ero}
\put(600,66){\rot{62} \sx{5}{$u\!=\!3$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(680,90){\rot{-16} \sx{5}{$v\!=\!3$} \ero}
\put(680,24){\rot{-8} \sx{5}{$v\!=\!1$} \ero}
\put(210,154){\rot{78} \sx{5}{$v\!=\!0$} \ero} 
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm05map}}  \put(20,340){\sx{7}{$n\!=\!-0.5$}}  
\put(602,386){\rot{-61} \sx{5}{$u\!=\!2$} \ero}
\put(382,302){\rot{-41} \sx{5}{$u\!=\!1$} \ero}
\put(20,272){\rot{-6} \sx{4.5}{$u\!\approx\!0.318$} \ero}
\put(82,206){\rot{-3} \sx{5}{$u\!=\!0$} \ero}
\put(10,122){\sx{5}{\bf cut}}
\put(486,298){\rot{42} \sx{5}{$v\!=\!2$} \ero}
\put(680,190){\rot{19} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp02map}}  \put(20,340){\sx{7}{$n\!=\!0.2$}}  \LeLa %\DoLa
\put(353,224){\rot{60} \sx{5}{$u\!=\!-1$} \ero}
\put(452,224){\rot{66} \sx{5}{$u\!=\!0$} \ero}
\put(530,200){\rot{71} \sx{5}{$u\!=\!1$} \ero}
\put(602,174){\rot{73} \sx{5}{$u\!=\!2$} \ero}
\put(10,122){\sx{5}{\bf cut}}
\put(680,328){\rot{-15} \sx{5}{$v\!=\!5$} \ero}
\put(680,260){\rot{-11} \sx{5}{$v\!=\!4$} \ero}
\put(680,194){\rot{-10} \sx{5}{$v\!=\!3$} \ero}
\put(680,125){\rot{-7} \sx{5}{$v\!=\!2$} \ero}
\put(680,56){\rot{-3} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(200,240){\rot{-31} \sx{4.6}{$v\!\approx\!1.337$} \ero}
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm02map}}  \put(20,340){\sx{7}{$n\!=\!-0.2$}}  %\DoLa 
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\put(302,326){\rot{-66} \sx{5}{$u\!=\!0$} \ero}
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%\put(680,348){\rot{5} \sx{5}{$v\!=\!3$} \ero}
\put(680,268){\rot{9} \sx{5}{$v\!=\!2$} \ero}
\put(470,132){\rot{8} \sx{4.6}{$v\!\approx\!1.337$} \ero}
%\put(470,128){\rot{4.6} \sx{5}{$v\!\approx\!1.3372357$} \ero}
\put(680,124){\rot{4} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
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\end{picture}}\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp01map}}  \put(20,340){\sx{7}{$n\!=\!0.1$}}  \LeLa \DoLa
\put(340,228){\rot{74} \sx{5}{$u\!=\!-1$} \ero}
\put(438,228){\rot{79} \sx{5}{$u\!=\!0$} \ero}
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\put(614,200){\rot{83} \sx{5}{$u\!=\!2$} \ero}
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\put(680,318){\rot{-6} \sx{5}{$v\!=\!4$} \ero}
\put(680,237){\rot{-5} \sx{5}{$v\!=\!3$} \ero}
\put(680,154){\rot{-4} \sx{5}{$v\!=\!2$} \ero}
\put(680,72){\rot{-3} \sx{5}{$v\!=\!1$} \ero}

\put(114,210){\rot{-19} \sx{4.6}{$v\!\approx\!1.337$} \ero}
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm01map}}  \put(20,340){\sx{7}{$n\!=\!-0.1$}}  \DoLa 
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\put(680,224){\rot{4} \sx{5}{$v\!=\!2$} \ero}
\put(470,128){\rot{4} \sx{4.6}{$v\!\approx\!1.337$} \ero}
%\put(470,128){\rot{4.6} \sx{5}{$v\!\approx\!1.3372357$} \ero}
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\end{document}

References

↑ https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
http://mizugadro.mydns.jp/BOOK/202.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
↑ http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

Keywords

«Abel function», «Complex map», «Exponential», «fsexp.cin», «fslog.cin», «Iterate», «Superfunction», «Superfunctions», «Tetration»,

«Суперфунцкии»,

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[1] ttps://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
http://mizugadro.mydns.jp/BOOK/202.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

[2] ttps://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.

[3] ttp://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
+{{oq|Expitemap.jpg|Original file ‎(1,825 × 2,841 pixels, file size: 1.88 MB, MIME type: image/jpeg)}}
-[[Complex map]]s of iterates of natural exponent;
+Fig.15.5 at page 214 of book «[[Superfunctions]]»<ref>
-$u\!+\!\mathrm i v=\exp^n(x\!+\!\mathrm i y)$
+https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862 <br>
+https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3<br>
+http://mizugadro.mydns.jp/BOOK/202.pdf
+Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
+</ref>, 2020:
+[[Complex map]]s of iterates of natural exponent ([[exponential]]);
+\(\ u\!+\!\mathrm i v=\exp^n(x\!+\!\mathrm i y)\)
-Usage: this is figure 15.5 of the book [[Суперфункции]] (2014, In Russian) <ref>
+The maps appear also in the Russian version «[[Суперфункции]]» (2014) <ref>
 https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
-http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
 http://mizugadro.mydns.jp/BOOK/202.pdf
-Д.Кузнецов. Суперфункции. [[Lambert Academic Publishing]], 2014.
+Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
+</ref>, Рис.15.5, стр. 213.
-</ref>; the English version is in preparation in 2015.
+The two maps at the top refers to \(n\!=\!1\) and \(n\!=\!-1\).
-Routines to evaluate the non–integer iterates are described also in the [[Vladikavkaz Matehmatical Journal]]
+They correspond to
+\(\exp^1\!=\!\exp\) and
+\(\exp^{-1}\!=\!\ln\).
+For \(n=0.5\), the detailed map of the [[Square root of exponential]] is loaded as
+https://mizugadro.mydns.jp/t/index.php/File:QexpMap1263x1243.png
+Case \(n=0\) ([[identity function]]) is not presented as obvious.
+Routines to evaluate the non-integer iterates of [[exponential]] are described also in the [[Vladikavkaz Mathematical Journal]]
 <ref>
 http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
-D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
+D.Kouznetsov. Superexponential as special function. [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-45.
 </ref>.
-==Refereces==
-<references/>
+The iterates of [[exponential]] are expressed through the [[natural tetration]] \(\mathrm{tet}\) and  [[arctetration]] \(\mathrm{tet}^{-1}\!=\!\mathrm{ate}\):
+\[
+\exp^n(z)=\mathrm{tet}\big(n+\mathrm{ate}(z)\big)
+\]
+In this expression, number \(n\) of the iterate has no need to be integer.
+The [[C++]] implementation of functions tet and ate are loaded as [[fsexp.cin]] and [[fslog.cin]].<br>
-==[[C++]] generator of the First map==
+The titles of these articles may look not so good. Names [[tet.cin]] and [[ate.cin]] (not loaded yet)
+are reserved for the optimized versions of these routines. The performance of the primary approximations can be significantly improved. This improvement can be achieved by the cost of reduction of region of the overlapping of the ranges of their validities. This overlapping is necessary for the numerical testing of the primary approximations of these functions.
+==[[C++]] generator of the First map==
-As the codes to generate the maps are very similar, I load the only generator of the first map. Files [[ado.cin]],
+/*
+As the codes to generate the maps are very similar, I load the only generator of the first map.<br>
+Files [[ado.cin]],
 [[conto.cin]],
 [[fsexp.cin]],
 [[fslog.cin]]
-should be loaded in order to compile the code below.
+should be loaded in order to compile the code below */
+<pre>
-<poem><nomathjax><nowiki>
 #include <math.h>
  #include <stdio.h>
@@ Line 97: / Line 121: @@
        getchar(); system("killall Preview"); // For macintosh
  }
+</pre>
-</nowiki></nomathjax></poem>
 ==[[Latex]] combiner==
+<pre>
-<poem><nomathjax><nowiki>
 \documentclass[12pt]{article}
 \usepackage{graphicx}
@@ Line 301: / Line 325: @@
 \end{document}
+</pre>
-</nowiki></nomathjax></poem>
+==References==
+{{ref}}
+{{fer}}
+==Keywords==
+«[[Abel function]]»,
+«[[Complex map]]»,
+«[[Exponential]]»,
+«[[fsexp.cin]]»,
+«[[fslog.cin]]»,
+«[[Iterate]]»,
+«[[Superfunction]]»,
+«[[Superfunctions]]»,
+«[[Tetration]]»,
+«[[Суперфунцкии]]»,
 [[Category:Abel function]]
 [[Category:Abelfunction]]
-[[Category:Agreement]]
 [[Category:Arctetration]]
 [[Category:Book]]
@@ Line 312: / Line 351: @@
 [[Category:Complex map]]
 [[Category:Exp]]
+[[Category:Exponential]]
 [[Category:Generator]]
 [[Category:Iterate]]
@@ Line 317: / Line 357: @@
 [[Category:Natural tetration]]
 [[Category:Superfunction]]
+[[Category:Superfunctions]]
 [[Category:Tetration]]