12/29/2010, 10:15 PM
(12/26/2010, 12:28 PM)sheldonison Wrote:(12/26/2010, 03:33 AM)JmsNxn Wrote: Sorry, you misunderstood me again. I meant precisely, does anybody know how to reproduce this graph? I have no means of evaluating \( \text{sexp}(\text{slog}(z)+0.5) \); although I know this is the derivation. I only know how to evaluate the linear approximation of tetration.Here are the sexp/slog Taylor series, for base e. Also, I updated my kneser.gp program which is at this link, http://math.eretrandre.org/tetrationforu...61#pid5461, which generates these Taylor series. You can also use kneser.gp to calculate sexp(slog(z)+0.5) directly.
- Sheldon
And here is the code to use kneser.gp to calculate and graph the partial iterates of exp(z).Code:gp kneser
init(exp(1));loop;
gp > sexptaylor(0,1)
sexp taylor series; first 60 terms of tseries[1..200] centered at 0
a0= 1.0000000000000000000000000000000
a1= 1.0917673512583209918013845500272
a2= 0.27148321290169459533170668362355
a3= 0.21245324817625628430896763774095
a4= 0.069540376139987373728674232707469
a5= 0.044291952090473304406440344385515
a6= 0.014736742096389391152096286915534
a7= 0.0086687818172252603663803925296399
a8= 0.0027964793983854596948259913011496
a9= 0.0016106312905842720721626451640261
a10= 0.00048992723148437733469866722583246
a11= 0.00028818107115404581134526404129648
a12= 0.000080094612538543333444273583010016
a13= 0.000050291141793805403694590114624202
a14= 0.000012183790344900091616191711098597
a15= 0.0000086655336673815746852458045540978
a16= 0.0000016877823193175389917890093175604
a17= 0.0000014932532485734925810665044317369
a18= 0.00000019876076420492745531981897951140
a19= 0.00000026086735600432637316458216085820
a20= 0.000000014709954142541901861412188187970
a21= 0.000000046834497327413506255093709923943
a22= -0.0000000015492416655467695218054651870829
a23= 0.0000000087415107813509359129925581144524
a24= -0.0000000011257873101030623175751345157011
a25= 0.0000000017079592672707284125656087892017
a26= -0.00000000037785831549229851764921434196894
a27= 0.00000000034957787651102163178731455708257
a28= -1.0537701234450015066294258142768 E-10
a29= 7.4590971476075052807322830644671 E-11
a30= -2.7175982065777348693298776352259 E-11
a31= 1.6460766106614471303885088308238 E-11
a32= -6.7418731524050529991474520798286 E-12
a33= 3.7253287233194685443170838697249 E-12
a34= -1.6390873267935902234582009653644 E-12
a35= 8.5836383113585680604886402833877 E-13
a36= -3.9437387391053843135795647434696 E-13
a37= 2.0025231280218870558935548339772 E-13
a38= -9.4419622429240650237152184959512 E-14
a39= 4.7120547458493713408175827120700 E-14
a40= -2.2562918820355970800425381378600 E-14
a41= 1.1154688506165369962926597065500 E-14
a42= -5.3907455570163504918451417705367 E-15
a43= 2.6521584915166818728187694206743 E-15
a44= -1.2889107655445536819358062274249 E-15
a45= 6.3266785019566604530407917519222 E-16
a46= -3.0854571504923359889458256800499 E-16
a47= 1.5131767717827405272874866193988 E-16
a48= -7.3965341370947514333107062095370 E-17
a49= 3.6269876710541876050990527120989 E-17
a50= -1.7757255986762984037825938163249 E-17
a51= 8.7098795443960546454574902166094 E-18
a52= -4.2692892823391563287091424171290 E-18
a53= 2.0950441625755281070058924218234 E-18
a54= -1.0278837092822587805532817556067 E-18
a55= 5.0468242474381764972455414919753 E-19
a56= -2.4780505958215523691539988885745 E-19
a57= 1.2173942030393316177075997930601 E-19
a58= -5.9816486323037832590313700332784 E-20
a59= 2.9402643445138963615125271861131 E-20
a60= -1.4455835436201860274824974353054 E-20
gp > slogtaylor(1,1)
slog taylor series; first 60 terms of tseries[1..200] centered at 1
a0= -1.3211559203569863866408422146826 E-34
a1= 0.91594605649953339394967263891032
a2= -0.20861842957759365309576465000989
a3= -0.054504006302093270028589453382969
a4= 0.071349419252730269716631593234174
a5= -0.020043873744376760638453315399497
a6= -0.011012580230372740346748219530571
a7= 0.012072683186448096707728724452800
a8= -0.0027292288076044037447174271067553
a9= -0.0026990531915602911438309647235647
a10= 0.0024394150063169339650512526852975
a11= -0.00036220360857878535132208036771582
a12= -0.00070125921261952546456510916763662
a13= 0.00052782155380099718416847990844747
a14= -0.000029879435510142989122473880516767
a15= -0.00018614540433646047225533197352996
a16= 0.00011722843042131144751177289862535
a17= 0.0000061161967982264073849038853296138
a18= -0.000049743993090135554955691635136395
a19= 0.000026094634277301257580080305012250
a20= 0.0000048654815670336045257505975678708
a21= -0.000013283077295456002362901764458609
a22= 0.0000057111621795780366934296661752353
a23= 0.0000020762671875207797595077206113096
a24= -0.0000035278409692466314741751176961981
a25= 0.0000012020714375007390285059722441766
a26= 0.00000074571746323099303987606383144658
a27= -0.00000092862508571439940061006130948504
a28= 0.00000023465874543866520339409599359245
a29= 0.00000024659053052096666651441342093773
a30= -0.00000024148031961242423040077519753963
a31= 0.000000039061466265679074042659723779693
a32= 0.000000077572889507124525250385241747965
a33= -0.000000061814687014870752758332787031310
a34= 0.0000000039100108857877969870065442074287
a35= 0.000000023568196115262335493443291946494
a36= -0.000000015507578660338656674652844066426
a37= -0.00000000076229263765083370463283196427707
a38= 0.0000000069696387132592104596552441583513
a39= -0.0000000037893014093949110070473762372749
a40= -0.00000000070245686946320817762679707951249
a41= 0.0000000020145444724257235214103778777926
a42= -0.00000000089336951487981237787929222961794
a43= -0.00000000032437195210353221968920308296109
a44= 0.00000000057034359496647503849260689198276
a45= -1.9996894318482819745356933325858 E-10
a46= -1.2372461492260537846045917929634 E-10
a47= 1.5826663028631842784811130522054 E-10
a48= -4.1172731622280591325394461073375 E-11
a49= -4.2963913901171591795867096162253 E-11
a50= 4.3026753818680902848211179232567 E-11
a51= -7.2110741666729535248006399436873 E-12
a52= -1.4081206506543100177700768020041 E-11
a53= 1.1441656044932434328551679669122 E-11
a54= -7.7881272466486164199114107160390 E-13
a55= -4.4306138664140498272258672443228 E-12
a56= 2.9670956633392477687326191891963 E-12
a57= 1.3306029980035303996455595608531 E-13
a58= -1.3506072858887418616164485867249 E-12
a59= 7.4655910375599970604021120738082 E-13
a60= 1.3512020855621850616914541445110 E-13
Code:iter(x,c)={
local(y);
if (x<0, y=slog(exp(x))-1+c);
if (x>=0, y=slog(x)+c);
if (y<-2,return(-3));
if (y>3, return(3));
y=sexp(y);
if (y<-3, return(-3));
if (y>3, return(3));
return(y);
}
ploth(t=-3,3,[iter(t,-1.5),iter(t,-1),iter(t,-0.5),iter(t,0),iter(t,0.5),iter(t,1),iter(t,1.5)]);
Thank you. Just one final question, how did you generate this Taylor series? Is there a closed form expression?