# Tetration Forum

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I was wondering if anyone had the means by which to reproduce this graph? And also, why this is not the accepted extension for tetration of rational values?

Sorry to be a newb and ask so many questions
(12/25/2010, 06:07 PM)JmsNxn Wrote: [ -> ]

I was wondering if anyone had the means by which to reproduce this graph? And also, why this is not the accepted extension for tetration of rational values?

Sorry to be a newb and ask so many questions
I believe this is a graph of $\exp^{[c]}(z)$. So the line with c=0.5 would be the half iterate of the exp(z), which can be calculated as $\exp^{[0.5]}(z)=\text{sexp}(\text{slog}(z)+0.5)$. For integer values of c, the equations are simpler. $\exp^{[2]}(z)=\text{sexp}(\text{slog}(z)+2)=\exp(\exp(z))$. And $\exp^{[-1]}(z)=\log(z)$
(12/25/2010, 07:06 PM)sheldonison Wrote: [ -> ]I believe this is a graph of $\exp^{[c]}(z)$. So the line with c=0.5 would be the half iterate of the exp(z), which can be calculated as $\exp^{[0.5]}(z)=\text{sexp}(\text{slog}(z)+0.5)$. For integer values of c, the equations are simpler. $\exp^{[2]}(z)=\text{sexp}(\text{slog}(z)+2)=\exp(\exp(z))$. And $\exp^{[-1]}(z)=\log(z)$

Sorry, I should've been more specific. I meant to ask for a formula independent of tetration; I'm assuming there's a power series of some kind defining 0<= q<=1$\exp^{[q]}(x)$ Something that would reproduce this graph, since the linear model of tetration I've been using doesn't match up. Even a generalized power series for tetration would work, now that I think of it.
(12/25/2010, 07:53 PM)JmsNxn Wrote: [ -> ]Sorry, I should've been more specific. I meant to ask for a formula independent of tetration; I'm assuming there's a power series of some kind defining 0<= q<=1$\exp^{[q]}(x)$ Something that would reproduce this graph, since the linear model of tetration I've been using doesn't match up. Even a generalized power series for tetration would work, now that I think of it.
For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via $\text{sexp}(\text{slog}(z)+0.5)$. I'm unaware of any other way to generate the power series.
- Shel

(12/25/2010, 10:52 PM)sheldonison Wrote: [ -> ]For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via $\text{sexp}(\text{slog}(z)+0.5)$. I'm unaware of any other way to generate the power series.
- Shel

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.
(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
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
And here is the code to use kneser.gp to calculate and graph the partial iterates of exp(z).
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)]);
(12/25/2010, 10:52 PM)sheldonison Wrote: [ -> ]For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via $\text{sexp}(\text{slog}(z)+0.5)$. I'm unaware of any other way to generate the power series.
- Shel

you could also use my method since e > e^(1/2).

although its not directly a power series nor the best numerically method its simple and doesnt use slog or sexp.

regards

tommy1729
(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
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
And here is the code to use kneser.gp to calculate and graph the partial iterates of exp(z).
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?
(12/29/2010, 10:15 PM)JmsNxn Wrote: [ -> ]Thank you. Just one final question, how did you generate this Taylor series? Is there a closed form expression?

James, as far as I see this is a graph which was produced by Dimitri Kousnetzov, who also posted here in the forum (you may use this link to find all all posts of him ) and to a certain extend explained his method here. But there is also a published paper of him where he describes this in detail (I've never understood it, btw, because I seem to lack some basic knowledge about cauchy-integrals and riemann-mappings, but for a student of mathematics this may be completely familiar). I think his article is also in our (the tetration-forum's) database of literature (see the related message lit-ref-db in the forum)

And have a happy new year -
Gottfried
(12/30/2010, 12:50 PM)Gottfried Wrote: [ -> ]
(12/29/2010, 10:15 PM)JmsNxn Wrote: [ -> ]Thank you. Just one final question, how did you generate this Taylor series? Is there a closed form expression?
James, as far as I see this is a graph which was produced by Dimitri Kousnetzov
There is no known closed form for the Taylor series for tetration. I generated the Taylor series with the kneser.gp program I wrote. I also posted the mathematical equations behind the algorithm here, http://math.eretrandre.org/tetrationforu...hp?tid=487

The basic idea using base e here, where L is the fixed point such that $L=\exp(L)$, $L\approx 0.318+1.317i$, is that if
$f(z)=L+\delta$ and $f(z+1)=\exp(f(z))=L\exp(\delta) \approx L+L\delta$ and $f(z+2)=\exp(\exp(f(z))) \approx L+L^2\delta$ etc.

This can be used to develop a complex valued entire superfunction such that $\text{superf}(z+1)=\exp(\text{superf}(z))$ for all values of z. $\text{superf}(z) = \lim_{n \to \infty} \exp^{[n]}(L + L^{z-n})$
The problem is that the superf is complex valued, not real valued. A 1-cyclic mapping is used to convert this function to an analytic real valued tetration. The 1-cyclic theta mapping is equivalent to the Riemann mapping in Kneser's algorithm, although convergence is not proven.
http://math.eretrandre.org/tetrationforu...hp?tid=487

The Taylor series is generated via a unit circle Cauchy integral.
- Sheldon
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