Difference between revisions of "File:Shelima600.png"

From TORI
Jump to navigation Jump to search
($ -> \( ; pre; ref; Keywords)
m
 
Line 17: Line 17:
 
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">
 +
In the Russian version «[[Суперфункции]]» <ref name="br">
 
https://mizugadro.mydns.jp/BOOK/202.pdf
  
https://mizugadro.mydns.jp/BOOK/202.pdf
 
Д.Кузнетсов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
  
Д.Кузнетсов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.

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

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current06:14, 1 December 2018Thumbnail for version as of 06:14, 1 December 20181,693 × 531 (83 KB)Maintenance script (talk | contribs)Importing image file

You cannot overwrite this file.

The following page uses this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:Shelima600.png&oldid=46406"