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An explanation for this?
#1
[Image: Expc.jpg]

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 Tongue
Reply
#2
(12/25/2010, 06:07 PM)JmsNxn Wrote: [Image: Expc.jpg]

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 Tongue
I believe this is a graph of . So the line with c=0.5 would be the half iterate of the exp(z), which can be calculated as . For integer values of c, the equations are simpler. . And
Reply
#3
(12/25/2010, 07:06 PM)sheldonison Wrote: I believe this is a graph of . So the line with c=0.5 would be the half iterate of the exp(z), which can be calculated as . For integer values of c, the equations are simpler. . And

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 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.
Reply
#4
(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 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 . I'm unaware of any other way to generate the power series.
- Shel

Reply
#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 . 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 ; although I know this is the derivation. I only know how to evaluate the linear approximation of tetration.
Reply
#6
(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 ; 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)]);
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#7
(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 . 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
Reply
#8
(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 ; 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?
Reply
#9
(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
Gottfried Helms, Kassel
Reply
#10
(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 , , is that if
and and etc.

This can be used to develop a complex valued entire superfunction such that for all values of z.
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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