02/14/2016, 08:29 PM
(This post was last modified: 02/14/2016, 09:16 PM by sheldonison.)
(02/14/2016, 07:31 PM)marraco Wrote:(02/14/2016, 07:55 AM)sheldonison Wrote: Those series are for the base not the exponent! That's why they needed to be generated with tetcomplex rather than kneser; because getting such amazingly good analytic convergence for such a series requires complex base tetration, with results for n bases, equally spaced around a circle centered on base=2. fatou.gp (with small code updates) could also generate such results. To evaluate those two referenced Taylor series, substitute x=(b-2), to evaluate the Taylor series.
...I'm royally confused now.
the coefficients in the last code tag of this post are the Taylor series coefficients of the function \( \\[15pt]
{f_2(x)\,=\,^{-0.5}(x+2)} \) ?
Then it should be
\( \\[15pt]
{f_{2\,(0)}\,=\, ^{-0.5}(2)=0.544764121459557...} \)
\( \\[15pt]
{f_{2\,(1)}\,=\, ^{-0.5}(3)=0.486648923935738...} \)
\( \\[15pt]
{f_{2\,(2)}\,=\, ^{-0.5}(4)=0.457213238216139...} \)
but this code gives
Code:\p 100
/*coefficients of Taylor around 2 of ^{-0.5}x*/
a=[0.5447641214595567339801218858257244685854,-0.09026490293475114180982800726025252487179,0.05334642698935378617403396491528890594804,-0.03638190492562309183765608353362070821840,0.02665589484943122254265742189263438424835,-0.02047608577133435850738520805893632252252,0.01628939391559684527389871185757624228228,-0.01331802035638468229849633176805710250959,0.01113080347039454404398917618932270486539,-0.009471945601741301301799666960159500493414,0.008181870472918983418952481797363865140773,-0.007156971109633091475785436209879906176635,0.006327698270413005651257016844418549882893,-0.005646005057506155565996841622134687059648,0.005077852297548008377590502397935756807242,-0.004598579564709383679003395147264261288216,0.004189960720720897899813797031566114828076,-0.003838281581489355718364575037825080832826,0.003533055202798846826754080155559369484869,-0.003266144873505376098605478730250127897586,0.003031153706186763516973507099991793317656,-0.002822992160610217843850530938287773588695,0.002637566583362648226391359841543536218954,-0.002471551499838161939220088380143251371568,0.002322220812129558330686955182616565785365,-0.002187321052201244713039707812416923885954,0.002064975080105974232286030081964966694861,-0.001953608108640999822645409586956590179027,0.001851890298539576237491912671987439471748,-0.001758691790434811807511081574014212915353,0.001673047168806168273884584564550319035826,-0.001594127148975512837191984637091772279620,0.001521215846034519175841930450931000214014,-0.001453692394304569109742037483974061901089,0.001391015984731342378663288972983370893753,-0.001332713607717317327426366538502721958881,0.001278369952559800466677607874323180950794,-0.001227619037446880124253911053592339307366,0.001180137236855264191223682764928143626889,-0.001135637444029102801867724419804092337925,0.001093864160641107689052187995385697434389,-0.001054589347847405322195327651270883669674,0.001017608905751088252280461602251417505390,-0.0009827396740070429313553656082778187553405,0.0009498168665858747234148639733725614346048,-0.0009186918698079533934930814875314534697736,0.0008892303455966866573662840017051026789547,-0.0008613105921955727892242590292733293823957,0.0008348221228916550711398042195621478636400,-0.0008096644300085595816921514966352297536765,0.0007857459069005338594524734325749675459648,-0.0007629829051481069438849922812669102687654,0.0007412989078245228555308404776565708380003,-0.0007206238027261275601524070871984237792707,0.0007008932419630012770449395923516096144246,-0.0006820480763866325438818551382445843370545,0.0006640338550679383137667663988829019940730,-0.0006468003814946490331294868382872885969872,0.0006303013193831178111678644623963748370560,-0.0006144938420376038744680011450272121474617,0.0005993383200741946673108758684076009577129,-0.0005847980430851115584178527733490186488280,0.0005708389714760195065962019338923580583096,-0.0005574295152845303003325133006371836953863,0.0005445403373002463827629604348689289744143,-0.0005321441782716469402918017155794455019026,0.0005202157024181592772390603384936993523783,-0.0005087313618820156701812678182987572351923,0.0004976692791697413648392248040346423784153,-0.0004870091470646722871348673605764087543724,0.0004767321459596961918548061350965022366495,-0.0004668208790873150944172435667760750155470,0.0004572593267416065803546985077871002284578,-0.0004480328213309398865880156803821028719749,0.0004391280460191844112860195774985849996584,-0.0004305330608687577109307488887504560310821,0.0004222373618726151852083171686555860996115,-0.0004142319801615488633254732943499557223787,0.0004065096311401751693759190311085411363661,-0.0003990649265288728503867805273227150792807,0.0003918946665218884447060007230972331199284,-0.0003849982348510417760694970639142865634935,0.0003783781269217428875539830317674205408416,-0.0003720406509700054967447945428476416910439,0.0003659968551918890293717641839010994444982,-0.0003602637511201750326035146298122448667960,0.0003548659266520138706765657962715630833711,-0.0003498376730740476659615080146624803134097,0.0003452257919088022517001222379805220095853,-0.0003410933031098378510324434621473247433045,0.0003375243510812609314744505648626776451409,-0.0003346307060211903759381304723940593677783,0.0003325603945037557882292885310966766766416,-0.0003315091777419398624384857779237117250571,0.0003317358460153907833479388101334190239424,-0.0003335826371421790840947235619676003126446,0.0003375025483293688889372157683210232309158,-0.0003440959391986930633575807516000693209207,0.0003541596811015852018702302245010936694298,-0.0003687532792553722994412436565871821184048,0.0003892879974033369355506985563538084697749,-0.0004176472122482091920832394022681079546693,0.0004563492419325118494433069912923916819317,-0.0005087680413650084606119114211931555096770,0.0005794328703416784956152178590887291116773,-0.0006744359202315312364732927380874961661109,0.0008019877695049482375578463387186199880001,-0.0009731755956181394060092634776613016580966,0.00120299993096644682942613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/*Taylor series around 2 of ^{-0.5}x*/
f2(x)=sum(n=0,150,a[n+1]/n!*x^n)
\( \\[15pt]
{f_{2\,(0)}\,=\, 0.5447641214595567...} \)
\( \\[15pt]
{f_{2\,(1)}\,=\, 0.4760690431769200...} \)
\( \\[15pt]
{f_{2\,(2)}\,=\, 0.4358972773362133...} \)
... and why textcomplex.gp "generates tetration for arbitrary bases" instead of "arbitrary exponents"? Sounds like the same as knesser and fatou (the base is constant, and the exponent is the variable)
Every infinite analytic Taylor series has a radius of convergence, which depends on the nearest singularity in the compplex plane. For the sexp_b(-0.5) the nearest singularity is for base b=eta, so the radius of convergence is (2-eta)~=0.555332. It turns out the singularity at eta is remarkably mild, so as I remember, you get good results even for b=2.6; evaluating with x=0.6. In fact, if you read the post, analyzing the Taylor series coefficients, they appear at first to be more limited by the singularity at x=1, then at etaB. But x=0.553322 is the theoretical limit of the convergence due to the nearest singularity. Those are the rules of analytic functions in the complex plane. I could post code to generate the half iterate centered at b=3, using fatou.gp, which would work for bases between eta and 4.6.
The process is fairly simple. Pick n bases, equally spaced around a circle in the complex plane around the base in question. Since complex base tetration is an analytic function, you get really good convergence, out to the radius of the sample points that were chosen, using Fourier/Cauchy to generate the Taylor series coefficients.
- Sheldon
