Difference between revisions of "File:Shelima600.png"

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{{oq|Shelima600.png|Original file ‎(1,693 × 531 pixels, file size: 83 KB, MIME type: image/png)|340}}
⚫
Explicit plot of real and imaginary part of [[tetration to Sheldon base]] of imaginary argument.
  
   
  +
Bottom part of
⚫
$b=1.52598338517 + 0.0178411853321~ \mathrm{i}$.
 
  +
Fig.18.3 at page 250 of book «[[Superfunctions]]»
  +
<ref name="be">
  ⚫
https://mizugadro.mydns.jp/BOOK/468.pdf
  ⚫
D.Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  ⚫
</ref>, 2020.
   
  ⚫
Explicit plot of real and imaginary part of [[tetration to Sheldon base]] \(b\) of imaginary argument.
⚫
The blue curve shows $y=\Re\Big( \mathrm{tet}_b(\mathrm i x)\Big)$
  
  +
Here
  +
\[
  ⚫
b=1.52598338517 + 0.0178411853321~ \mathrm{i}
  +
\]
  ⚫
The blue curve shows \(y=\Re\Big( \mathrm{tet}_b(\mathrm i x)\Big)\)
   
−
The red curve shows $y=\Im\Big( \mathrm{tet}_b(\mathrm i x)\Big)$
 +
The red curve shows \(y=\Im\Big( \mathrm{tet}_b(\mathrm i x)\Big)\)
   
  ⚫
In the Russian version «[[Суперфункции]]» <ref name="br">
−
==Usage==
  
  +
https://mizugadro.mydns.jp/BOOK/202.pdf
−
The image is prototype of figure 18.5 of the English version of the book [[Superfunctions]]
  
  +
Д.Кузнетсов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
⚫
<ref>
  
  +
</ref>, 2014, this picture is absent.
⚫
http://mizugadro.mydns.jp/BOOK/437.pdf
  
⚫
D.Kouznetsov. Superfunctions. 2015.
  
⚫
</ref>.
  
−  
⚫
In the first Russian version [[Суперфункции]] of the book, this image is absent.
  
−  
⚫
==References==
  
−
<references/>
  
   
 
==Table of values ==
  
==Table of values ==
 
<poem>
  
<poem>
−
$~ ~x~ ~$ $~ ~\Re(\mathrm{tet}_b(\mathrm i x))~ ~$ $~ ~\Im(\mathrm{tet}_b(\mathrm i x))$
 +
\(~ ~x~ ~\) \(~ ~\Re(\mathrm{tet}_b(\mathrm i x))~ ~\) \(~ ~\Im(\mathrm{tet}_b(\mathrm i x))\)
 
-10.00 2.2202212609 -1.3393771522
  
-10.00 2.2202212609 -1.3393771522
 
-9.90 2.2196731381 -1.3388553660
  
-9.90 2.2196731381 -1.3388553660
Line 226: Line 229:
   
 
==[[C++]] generator of curves==
  
==[[C++]] generator of curves==
  ⚫
/*Files
−  
⚫
Files
  
 
[[GLxw2048.inc]] ,
  
[[GLxw2048.inc]] ,
 
[[TetSheldonIma.inc]] ,
  
[[TetSheldonIma.inc]] ,
Line 233: Line 235:
 
[[conto.cin]] ,
  
[[conto.cin]] ,
 
[[filog.cin]]
  
[[filog.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 <math.h>
 
#include <stdio.h>
  
#include <stdio.h>
Line 299: Line 300:
 
getchar(); system("killall Preview");
  
getchar(); system("killall Preview");
 
}
  
}
  +
</pre>
−  
−
</nowiki></nomathjax></poem>
  
−  
 
==[[Latex]] generator of labels==
  
==[[Latex]] generator of labels==
  +
<pre>
−  
−
<poem><nomathjax><nowiki>
  
 
\documentclass[12pt]{article}
  
\documentclass[12pt]{article}
 
\usepackage{geometry}
  
\usepackage{geometry}
Line 341: Line 339:
 
\end{picture}
  
\end{picture}
 
\end{document}
  
\end{document}
  +
</pre>
−
</nowiki></nomathjax></poem>
  
  ⚫
==References==
  ⚫
{{ref}}
   
  +
{{fer}}
  +
==Keywords==
  +
«[[]]»,
  +
«[[Explicit plot]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
«[[Tetration]]»,
  +
«[[Tetration to Sheldon base]]»,
  +
«[[]]»,
   
 
[[Category:Book]]
  
[[Category:Book]]
Line 352: Line 361:
 
[[Category:Tetration to Sheldon base]]
  
[[Category:Tetration to Sheldon base]]
 
[[Category:Sheldon base]]
  
[[Category:Sheldon base]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]

Latest revision as of 22:25, 3 January 2026


Bottom part of Fig.18.3 at page 250 of book «Superfunctions» [1], 2020.

Explicit plot of real and imaginary part of tetration to Sheldon base \(b\) of imaginary argument. Here \[ b=1.52598338517 + 0.0178411853321~ \mathrm{i} \] The blue curve shows \(y=\Re\Big( \mathrm{tet}_b(\mathrm i x)\Big)\)

The red curve shows \(y=\Im\Big( \mathrm{tet}_b(\mathrm i x)\Big)\)

In the Russian version «Суперфункции» [2], 2014, this picture is absent.

Table of values

\(~ ~x~ ~\) \(~ ~\Re(\mathrm{tet}_b(\mathrm i x))~ ~\) \(~ ~\Im(\mathrm{tet}_b(\mathrm i x))\)
-10.00 2.2202212609 -1.3393771522
 -9.90 2.2196731381 -1.3388553660
 -9.80 2.2190910829 -1.3383114251
 -9.70 2.2184731185 -1.3377444826
 -9.60 2.2178171599 -1.3371536640
 -9.50 2.2171210078 -1.3365380660
 -9.40 2.2163823431 -1.3358967563
 -9.30 2.2155987208 -1.3352287720
 -9.20 2.2147675635 -1.3345331194
 -9.10 2.2138861545 -1.3338087726
 -9.00 2.2129516317 -1.3330546724
 -8.90 2.2119609791 -1.3322697257
 -8.80 2.2109110202 -1.3314528034
 -8.70 2.2097984096 -1.3306027402
 -8.60 2.2086196248 -1.3297183321
 -8.50 2.2073709574 -1.3287983354
 -8.40 2.2060485042 -1.3278414651
 -8.30 2.2046481579 -1.3268463926
 -8.20 2.2031655968 -1.3258117437
 -8.10 2.2015962752 -1.3247360964
 -8.00 2.1999354119 -1.3236179782
 -7.90 2.1981779800 -1.3224558632
 -7.80 2.1963186945 -1.3212481688
 -7.70 2.1943520007 -1.3199932521
 -7.60 2.1922720612 -1.3186894058
 -7.50 2.1900727435 -1.3173348536
 -7.40 2.1877476059 -1.3159277450
 -7.30 2.1852898835 -1.3144661496
 -7.20 2.1826924737 -1.3129480506
 -7.10 2.1799479207 -1.3113713370
 -7.00 2.1770484004 -1.3097337961
 -6.90 2.1739857032 -1.3080331037
 -6.80 2.1707512180 -1.3062668134
 -6.70 2.1673359142 -1.3044323456
 -6.60 2.1637303238 -1.3025269736
 -6.50 2.1599245231 -1.3005478090
 -6.40 2.1559081133 -1.2984917849
 -6.30 2.1516702011 -1.2963556371
 -6.20 2.1471993787 -1.2941358833
 -6.10 2.1424837030 -1.2918287992
 -6.00 2.1375106749 -1.2894303921
 -5.90 2.1322672181 -1.2869363712
 -5.80 2.1267396576 -1.2843421145
 -5.70 2.1209136985 -1.2816426317
 -5.60 2.1147744042 -1.2788325222
 -5.50 2.1083061753 -1.2759059295
 -5.40 2.1014927291 -1.2728564888
 -5.30 2.0943170796 -1.2696772695
 -5.20 2.0867615183 -1.2663607109
 -5.10 2.0788075973 -1.2628985502
 -5.00 2.0704361133 -1.2592817434
 -4.90 2.0616270956 -1.2555003763
 -4.80 2.0523597956 -1.2515435664
 -4.70 2.0426126821 -1.2473993541
 -4.60 2.0323634408 -1.2430545817
 -4.50 2.0215889804 -1.2384947601
 -4.40 2.0102654464 -1.2337039220
 -4.30 1.9983682448 -1.2286644588
 -4.20 1.9858720768 -1.2233569428
 -4.10 1.9727509879 -1.2177599308
 -4.00 1.9589784331 -1.2118497494
 -3.90 1.9445273631 -1.2056002612
 -3.80 1.9293703334 -1.1989826096
 -3.70 1.9134796423 -1.1919649425
 -3.60 1.8968275023 -1.1845121140
 -3.50 1.8793862518 -1.1765853645
 -3.40 1.8611286122 -1.1681419794
 -3.30 1.8420280011 -1.1591349277
 -3.20 1.8220589080 -1.1495124836
 -3.10 1.8011973445 -1.1392178346
 -3.00 1.7794213797 -1.1281886816
 -2.90 1.7567117731 -1.1163568397
 -2.80 1.7330527191 -1.1036478499
 -2.70 1.7084327173 -1.0899806167
 -2.60 1.6828455826 -1.0752670901
 -2.50 1.6562916111 -1.0594120161
 -2.40 1.6287789134 -1.0423127871
 -2.30 1.6003249281 -1.0238594279
 -2.20 1.5709581198 -1.0039347646
 -2.10 1.5407198646 -0.9824148312
 -2.00 1.5096665094 -0.9591695767
 -1.90 1.4778715856 -0.9340639471
 -1.80 1.4454281312 -0.9069594235
 -1.70 1.4124510594 -0.8777161012
 -1.60 1.3790794751 -0.8461953971
 -1.50 1.3454788118 -0.8122634650
 -1.40 1.3118426170 -0.7757953788
 -1.30 1.2783937743 -0.7366801138
 -1.20 1.2453849119 -0.6948263040
 -1.10 1.2130977120 -0.6501686844
 -1.00 1.1818408268 -0.6026750337
 -0.90 1.1519461162 -0.5523533237
 -0.80 1.1237629810 -0.4992586603
 -0.70 1.0976506656 -0.4434994833
 -0.60 1.0739685645 -0.3852424012
 -0.50 1.0530647752 -0.3247149921
 -0.40 1.0352633815 -0.2622059382
 -0.30 1.0208511970 -0.1980619898
 -0.20 1.0100649016 -0.1326814891
 -0.10 1.0030796228 -0.0665045046
  0.00 1.0000000000 0.0000000000
  0.10 1.0008546120 0.0663491799
  0.20 1.0055943462 0.1320624338
  0.30 1.0140948929 0.1966774148
  0.40 1.0261631142 0.2597647135
  0.50 1.0415466506 0.3209401766
  0.60 1.0599458447 0.3798741265
  0.70 1.0810269204 0.4362971165
  0.80 1.1044353661 0.4900022286
  0.90 1.1298086052 0.5408442355
  1.00 1.1567872534 0.5887361692
  1.10 1.1850245145 0.6336439551
  1.20 1.2141935082 0.6755797922
  1.30 1.2439925335 0.7145949050
  1.40 1.2741484179 0.7507721909
  1.50 1.3044182070 0.7842191660
  1.60 1.3345894889 0.8150614867
  1.70 1.3644796637 0.8434372142
  1.80 1.3939344425 0.8694918991
  1.90 1.4228258303 0.8933744915
  2.00 1.4510498021 0.9152340382
  2.10 1.4785238390 0.9352170954
  2.20 1.5051844494 0.9534657700
  2.30 1.5309847655 0.9701163005
  2.40 1.5558922770 0.9852980850
  2.50 1.5798867375 0.9991330757
  2.60 1.6029582648 1.0117354642
  2.70 1.6251056396 1.0232115932
  2.80 1.6463348018 1.0336600413
  2.90 1.6666575325 1.0431718336
  3.00 1.6860903115 1.0518307431
  3.10 1.7046533328 1.0597136520
  3.20 1.7223696637 1.0668909512
  3.30 1.7392645315 1.0734269577
  3.40 1.7553647215 1.0793803392
  3.50 1.7706980741 1.0848045336
  3.60 1.7852930661 1.0897481563
  3.70 1.7991784654 1.0942553916
  3.80 1.8123830487 1.0983663636
  3.90 1.8249353724 1.1021174851
  4.00 1.8368635897 1.1055417832
  4.10 1.8481953050 1.1086692018
  4.20 1.8589574618 1.1115268814
  4.30 1.8691762567 1.1141394158
  4.40 1.8788770769 1.1165290889
  4.50 1.8880844555 1.1187160898
  4.60 1.8968220432 1.1207187105
  4.70 1.9051125920 1.1225535247
  4.80 1.9129779496 1.1242355514
  4.90 1.9204390625 1.1257784031
  5.00 1.9275159849 1.1271944198
  5.10 1.9342278938 1.1284947908
  5.20 1.9405931079 1.1296896652
  5.30 1.9466291099 1.1307882509
  5.40 1.9523525708 1.1317989054
  5.50 1.9577793759 1.1327292168
  5.60 1.9629246530 1.1335860773
  5.70 1.9678027996 1.1343757500
  5.80 1.9724275118 1.1351039284
  5.90 1.9768118128 1.1357757907
  6.00 1.9809680809 1.1363960488
  6.10 1.9849080779 1.1369689923
  6.20 1.9886429758 1.1374985283
  6.30 1.9921833839 1.1379882174
  6.40 1.9955393749 1.1384413061
  6.50 1.9987205098 1.1388607560
  6.60 2.0017358620 1.1392492708
  6.70 2.0045940414 1.1396093194
  6.80 2.0073032162 1.1399431581
  6.90 2.0098711350 1.1402528501
  7.00 2.0123051475 1.1405402826
  7.10 2.0146122241 1.1408071836
  7.20 2.0167989751 1.1410551358
  7.30 2.0188716688 1.1412855895
  7.40 2.0208362487 1.1414998751
  7.50 2.0226983501 1.1416992133
  7.60 2.0244633157 1.1418847248
  7.70 2.0261362105 1.1420574393
  7.80 2.0277218365 1.1422183034
  7.90 2.0292247454 1.1423681880
  8.00 2.0306492523 1.1425078945
  8.10 2.0319994475 1.1426381610
  8.20 2.0332792080 1.1427596676
  8.30 2.0344922089 1.1428730416
  8.40 2.0356419337 1.1429788616
  8.50 2.0367316841 1.1430776619
  8.60 2.0377645897 1.1431699357
  8.70 2.0387436168 1.1432561392
  8.80 2.0396715769 1.1433366941
  8.90 2.0405511348 1.1434119905
  9.00 2.0413848162 1.1434823898
  9.10 2.0421750149 1.1435482264
  9.20 2.0429239999 1.1436098105
  9.30 2.0436339216 1.1436674297
  9.40 2.0443068184 1.1437213507
  9.50 2.0449446220 1.1437718210
  9.60 2.0455491634 1.1438190707
  9.70 2.0461221781 1.1438633134
  9.80 2.0466653107 1.1439047477
  9.90 2.0471801203 1.1439435581
10.00 2.0476680845 1.1439799167

C++ generator of curves

/*Files GLxw2048.inc , TetSheldonIma.inc , ado.cin , conto.cin , filog.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 "filog.cin"
z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
z_type a=log(b);
z_type Zo=Filog(a);
z_type Zc=conj(Filog(conj(a)));
DB A=32.;
z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
#include "GLxw2048.inc"
int K=2048;
//#include "ima6.inc"
#include "TetSheldonIma.inc"
z_type E[2048],G[2048];
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
c=0.;
// z+=z_type(0.1196573712872846, 0.1299776198056910);
z+=z_type( 0.1196591376539 , 0.1299777213955 );
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));} 
 cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
 cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
 c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
 return c;}

int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;

// FILE *o;o=fopen("tetsheldonmap.eps","w");ado(o,602,202);
FILE *o;
o=fopen("04.eps","w");ado(o,202,62);
 fprintf(o,"101 31 translate\n 10 10 scale\n");

 for(m=-10;m<11;m++){if(m==0){M(m,-3.2)L(m,3.2)} else{M(m,-3)L(m,3)}}
 for(n=-3;n<4;n++){ M( -10,n)L(10,n)}
 fprintf(o,".008 W 0 0 0 RGB S\n");

DO(m,201){x=.1*(m-100); z=z_type(0,x);  c=tetb(z); 
        p=Re(c); q=Im(c); 
        y=p; if(m==0) M(x,y) else L(x,y) 
        printf("%6.2lf %14.10lf %14.10lf\n",x,p,q); 
        }
 fprintf(o,".04 W 0 0 1 RGB S\n");

DO(m,201){x=.1*(m-100); z=z_type(0,x);  c=tetb(z); 
        p=Re(c); q=Im(c); 
        y=q; if(m==0) M(x,y) else L(x,y) }
 fprintf(o,".04 W 1 0 0 RGB S\n");


 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf 04.eps"); 
        system( "open 04.pdf");
        getchar(); system("killall Preview");
 }

Latex generator of labels

 \documentclass[12pt]{article}
 \usepackage{geometry}
 \paperwidth 204pt
 \paperheight 64pt
 %\textwidth 700pt
 \usepackage{graphics}
 \newcommand \sx \scalebox
 \newcommand \ing \includegraphics
\parindent 0pt
\topmargin -108pt
\oddsidemargin -72pt
\begin{document}
\begin{picture}(602,62)
\put(0,0){\ing{04}}
\put(97,59){\sx{.6}{$y$}}
\put(97,49){\sx{.6}{$2$}}
\put(97,39){\sx{.6}{$1$}}
\put(97,29){\sx{.6}{$0$}}
\put(92,19){\sx{.6}{$-1$}}
\put(92, 9){\sx{.6}{$-2$}}

\put(15,25){\sx{.6}{$-8$}}
\put(35,25){\sx{.6}{$-6$}}
\put(55,25){\sx{.6}{$-4$}}
\put(75,25){\sx{.6}{$-2$}}
\put(100,25){\sx{.6}{$0$}}
\put(120,25){\sx{.6}{$2$}}
\put(140,25){\sx{.6}{$4$}}
\put(160,25){\sx{.6}{$6$}}
\put(180,25){\sx{.6}{$8$}}
\put(199,25){\sx{.6}{$x$}}

\put(10,56){\sx{.7}{$y\!=\! \Re\big(\mathrm{tet}_b(\mathrm i x)\big)$}}
\put(10,11){\sx{.7}{$y\!=\! \Im\big(\mathrm{tet}_b(\mathrm i x)\big)$}}
\end{picture}
\end{document}

References

Keywords

«[[]]», «Explicit plot», «Superfunction», «Superfunctions», «Tetration», «Tetration to Sheldon base», «[[]]»,

File history

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