File:Shelima600.png

Explicit plot of real and imaginary part of tetration to Sheldon base of imaginary argument.

$b=1.52598338517 + 0.0178411853321~ \mathrm{i}$.

The blue curve shows $y=\Re\Big( \mathrm{tet}_b(\mathrm i x)\Big)$

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

Usage
The image is prototype of figure 18.5 of the English version of the book Superfunctions .

In the first Russian version Суперфункции of the book, this image 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:

// using namespace std; typedef std::complex z_type; 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; int K=2048; //#include "ima6.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;}
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 5) include "filog.cin"
 * 1) include "GLxw2048.inc"
 * 1) include "TetSheldonIma.inc"

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}