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Tetration series for integer exponent. Can you find the pattern?
#11
(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 ?

Then it should be




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,-0.0009731755956181394060092634776613016580966,0.001202999930966446829426139160934525077288,-0.001511794692123179861160636690582499120690,0.001927175422690043740636959464774028693374,-0.002486716638299575521444734874764590083602,0.003241637115621436229181368952656278856890,-0.004261880730986434210647558463981630316890,0.005643132413944208166578890706551114196855,-0.007516521379651240969156586191270607737992,0.01006206163735266482138789454244816545884,-0.01352729756512357159032473435823294438533,0.01825320917728106084640001141407135014463,-0.02471025705364750094828467307169149418527,0.03354860917065776303709779278149423244864,-0.04566823068915574646795726336973581269876,0.06231683139161434741416351784301382413064,-0.08522693583797479716332856332368442457569,0.1168079697188183445926413360961134224671,-0.1604158141872473246671531628893015031994,0.2207315839610322539821107999865013181935,-0.3042945998433380627917194164914032453560,0.4202533179050196135850984244232386711566,-0.5814247303292241247446316577711695736830,0.8057908832077465638002237284863080618377,-1.118615564350797221844687119564496999668,1.555441935527913324823119191631251168563,-2.166343033000847467475898770333491396571,3.021956187494805723428091061670418388443,-4.222060471273729919015929034763285805797,5.907783486640862219163610751023466051097,-8.278993788941744624065220674384056537610,11.61911094227013680329706281281624602869,-16.33053766929996258807625048927333995682,22.98531974919989325688421392691882638831,-32.39766033022431437194340100803250018929,45.72783261862920895891344502067529674014,-64.63125037132339003279765074797075722636,91.47255488228885159381088483245314278778,-129.6334095048019025160129531044130526933,183.9554899644497666098851981405887199496,-261.3787224160197438528735946730321986942,371.8617901668770815153842203841958000090,-529.7111364131130428052370084930064042101,755.5016929935324645636711119283331834056]

/*Taylor series around 2 of ^{-0.5}x*/
f2(x)=sum(n=0,150,a[n+1]/n!*x^n)






... 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
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#12
(02/14/2016, 05:08 AM)marraco Wrote:
(02/14/2016, 03:43 AM)Gottfried Wrote:
That's great. You found that the tetration of the pascal matrix produces the bj sequences.
If we can iterate logaritms maybe we can figure what a tree of negative height is.

I wonder if we can generate the tetrated pascal with a series, and if that series matches the series for some real base. That base should be important.

If just we could calculate , we would get the bj for trees of 1/2 height.

Hi marraco -

the point of the logarithm L of the Pascalmatrix P and its non-invertibility is likely not the end of the story here. Although it is only one subdiagonal below the diagonal and the diagonal is empty/is zero this is the same situation as we face it with the fractional derivative, expressed in the matrix form. There the derivative-operator is exactly that subdiagonal-matrix. And the second-derivative-operator is its second power and so on. But anyway we can work with somehow "the inverse", namely the integration. That operator has the upper subdiagonal populated with values. One of the reasons for the possibility to step forwards is likely the fact of the infinite size of the matrices which allow sometimes operations which are impossible/contradictory with finite sized-matrices.

And finally, there is also the concept of fractional derivatives (in fact a complete handful of concepts) which indeed define a squareroot of that derivative-operator or subdiagonal-matrix. However it must be made working seriously and I feel a bit tired to dive into this -for me new- concept. I tried to become more familiar with it and, for instance got stuck with the question how to define the half-derivative of the zeta at zero (based on the Dirichlet-series representation) and this question is still open in mathoverflow... So because I also had not much time besides my teachings this still lingers around - and if you like that whole question more and like try experimentate with that matrix-operators in Pari/GP you can also get my set of routines (which can help to make the matrix-things better visible - for instance all the nice matrix-pictures in my article are made with the PariTTY-GUI for Pari/GP). (You better email me privately for this because I surely must explain in detail this and that and much and more...)

Gottfried
Gottfried Helms, Kassel
Reply
#13
(02/14/2016, 08:29 PM)sheldonison Wrote: 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.

I had a course in complex analysis, and error propagation, but I'm still chewing it.

Aside of being a bit rusted, error control is critical here. The smaller the radius of the circle, the smaller theoretical error, but fatou only gives a limited number of decimals, no matter how much I crank up precision in pari/gp, so if I reduce the radius too much, all decimals get truncated, and I lose all the information.
If I rise the radius, then the error comes from error propagation and singularites. No clue about what is the optimal radius.

also, I need a base on the real line, as close as possible to 1, but fatou doesn't calc on bases with negative imaginary parts, so I need to integrate over a semicircle; cutting the circle with the real line, and that complicates tracing the error.

Despite Sheldonison kind attempts to explain fatou.gp algorithm, it goes over my head, so I cannot tweak fatou for more decimals, and must treat it like a black box.
I have the result, but I do not yet know how to get it.
Reply
#14
(02/15/2016, 08:11 PM)marraco Wrote: Despite Sheldonison kind attempts to explain fatou.gp algorithm, it goes over my head, so I cannot tweak fatou for more decimals, and must treat it like a black box.

^^I found my problem with fatou.gp

because I cranked up precision, and calculation of tetrations took hours, I was saving the results on a file using the function write(path,sexp(z)), so I could reutilize his results multiple times.

But fatou's sexp() function is formatted to display only 15 decimals. Saving to disk with write() only saves 15 decimals, so it destroys all the hours of work.

I was believing that fatou was fundamentally limited to 15 decimals and the algorithm only showed meaningful digits...

Well, I don't really know, so I will ask on fatou's thread.

EDIT: oh, it actually have 15 digits of precision.
(07/10/2015, 08:58 PM)sheldonison Wrote: A 512x512 matrix gives results accurate to about 15-16 decimal digits for base(e), . The fatou.gp pari-program uses theta mapping to iteratively solve p(z), to dramatically improve convergence.
I have the result, but I do not yet know how to get it.
Reply
#15
(02/15/2016, 10:08 PM)marraco Wrote:
(02/15/2016, 08:11 PM)marraco Wrote: Despite Sheldonison kind attempts to explain fatou.gp algorithm, it goes over my head, so I cannot tweak fatou for more decimals, and must treat it like a black box.

^^I found my problem with fatou.gp

because I cranked up precision, and calculation of tetrations took hours, I was saving the results on a file using the function write(path,sexp(z)), so I could reutilize his results multiple times.

But fatou's sexp() function is formatted to display only 15 decimals. Saving to disk with write() only saves 15 decimals, so it destroys all the hours of work.

I was believing that fatou was fundamentally limited to 15 decimals and the algorithm only showed meaningful digits...

Well, I don't really know, so I will ask on fatou's thread.

EDIT: oh, it actually have 15 digits of precision.
(07/10/2015, 08:58 PM)sheldonison Wrote: A 512x512 matrix gives results accurate to about 15-16 decimal digits for base(e), . The fatou.gp pari-program uses theta mapping to iteratively solve p(z), to dramatically improve convergence.


fatou currently defaults to disabling the theta mapping when you're anywhere near the attracting/repelling boundary, the "shell thron" boundary, where the function is approximately indifferent. Without a theta mapping means precision is limited to about 15-16 decimal digits. Allowing a theta mapping actual works fine unless you're almost exactly on the indifferent boundary, which is rarely the case. So you need some patches/updates, for fatou.gp

I'm waiting a little while more before I update the fatou.gp in the computation thread, before deciding what defaults make the most sense, but here it is for now. fatou.gp update:
.gp   fatou.gp (Size: 46.67 KB / Downloads: 150)

Also, you need something to make a taylor series. Here's some additional code that samples 100 points centered on the real axis, and uses the conjugate of those 100 points in the negative half of the real axis, to accurately interpolate the half(b) function centered at b=3, with radius 1.5 accurate to about 32 decimal digits. It takes about 6 mintues to run. Most of the bases can be calculated in a few seconds, but the one closet to the Shel Thron boundary needs a couple of minutes. The theta mapping allows you to have exponential convergence, even if it does get slow down really badly near the Shel Thron boundary. Without the theta mapping, you only get x^4.5 or so convergence, so you need a lot more iterations and a lot more sample points, and then you still wind up with only 16 decimal digits unless you have an infinite amount of time.

Code:
/* in fatou.gp p 38 gives you about 33-35 decimal digits of usable precision */
\p 38
default(format,"g0.15");
/* ffunc returns the sexp(-0.5) for base z */
ffunc(z) = {local(y,y2,t2);
  quietmode=1;
  gettime();
  setdefaults();
  print1(z" ");
  y2=loop(log(log(z))+1); /* loop is like sexpinit(z) using loop(k) */
  y=sexp(-0.5);
  t2=gettime();
  print(y2[1]" "t2);
  return(y);
}
/* ftaylor generates a taylor series centered at w, radius r, samples/2 */
ftaylor( w,r,samples) = {
  local(rinv,s,t,x1,y,z,tot,t_est,tcrc,halfsamples,wtaylor,terms,s2);
  if (samples==0, samples=200);
  halfsamples=samples/2;
  terms = floor(samples*200/240);
  t_est    = vector (samples,i,0);
  tcrc     = vector (samples,i,0);
  if (r==0,r=1);
  rinv = 1/r;
  wtaylor=0;
  s2=samples/2;
  for(s=1, samples, x1=-1/(samples)+(s/halfsamples); tcrc[s]=exp(Pi*I*x1); );
  for (t=1,s2, t_est[t] = ffunc(w+r*tcrc[t]); );
  for (t=s2+1,samples, t_est[t] = conj(t_est[samples+1-t]));

  for (s=0,terms-1,
    tot=0;
    for (t=1,samples,
      tot=tot+t_est[t];
      t_est[t]=t_est[t]*conj(tcrc[t]);
    );
    tot=tot/samples;
    if (s>=1, tot=tot*(rinv)^s);
    wtaylor=wtaylor+tot*x^s;
  );
  wtaylor=precision(wtaylor,precis);
  return(real(wtaylor)); /* real valued; ignore imaginary part errors */
}
ft=ftaylor(3,1.45,200); /* centered at 3, radius=1.45, 100 sample points */
default(format,"g0.32");
prtpoly(ft,150,"xhalf");  /* print out accurate to 32 decimal digits */
- Sheldon
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#16
You are great, Sheldonison.

(I had lot of work, so I was unable to answer for like a week.)

I sampled 1000 points around the point r=1.1, with a radius R=.01.

this is the code:

Code:
\r fatou.gp
\p 100
/*Integration path over a semicircle*/

r=1.1            /*center of calculation*/
R=(r-1)/10       /*Radius of integration*/
Ndq=1000         /*number of angles*/
dq=2*Pi/N°dq     /*differential of angle*/


/*Calculation points/path z of integral*/
/*[z0,z1,z2,...,z(Ndq)=z0]*/ /*last point==first*/
z=vector(1+Ndq,n,  r+R*exp(I*(n-1)*dq)  )


/*(z-a) average over the integration interval*/
/*[0_1,1_2,2_3,...,Ndq_0]*/
z_a=vector(Ndq,n,  R*exp(I*(n*dq-dq/2))   )


/*differential vector*/
/*[d1_0,d2_1,d3_1,...,d0_Ndq]*/
dz=vector(Ndq,n,  dq*(z_a[n]*I) )

/*start integrand numerator*/
    /*calculation of tetrations*/
    TetZ = z*0  /*Same size as z, and initializated=0*/
    /*[Tz0,Tz1,Tz2,...,Tz(Ndq)=Tz0]*/ /*last point==first*/
    {
    for(n=1,1+Ndq,
       if(imag(z[n])<0,
          sexpinit(conj(z[n]));
          TetZ[n]=conj(sexp(1.5));
          ,
          sexpinit(z[n]);
          TetZ[n]=sexp(1.5);
        );
    )};
    
    /*Calculation of numerator of integrand*/
    /*Same size as z_a*/
    /*[I0_1,I1_2,I2_3,...,INdq_0]*/
    Integrand_num=vector(length(z_a),n, (TetZ[n]+TetZ[n+1])/2 *  dz[n]  )
/*end integrand numerator*/

/*Cauchy formula*/
jDerivative=vector(50,j,(j-1)!/(2*Pi*I)*sum(n=1,length(Integrand_num),Integrand_num[n]/z_a[n]^(j)))
(I need the Taylor centered at 1, so I cannot avoid calculating inside the shelltron region)

It took days to run, but I only got 7 converging derivatives (I know that the Taylor series (around r) should be equal to 1 when evaluated at 1, so I count the converged derivatives as the ones that give a result closest to the correct one.

I also was forced to pick only the real part of the coefficients.

Anyways, I got those bj:

[0.998577152804501, 2.89169703329950, -9.32293754538342, 410.790219668510, -8400.01108400142, 77233.5910778554, 1116744.84553185]

Unfortunately, it is not much better than the last attempt:
(02/13/2016, 06:07 AM)marraco Wrote:
Code:
%38 = [1.000152704474621237210741951681248733595287259524430467296, 2.861320494145353285475346429548911736678675196129994723805, -13.6475978846314412502384523327046630578, 756.92186972158813546692266102007258863]
maybe using a fourier series to interpolate fatou.gp results improve the integral.

I will attempt to implement a conjugate gradient on the red blue equation, use knesser.gp to get a very good first guess, and increase precision.
That also may lead nowhere, since the red blue eq is mined with local minima.


I have the result, but I do not yet know how to get it.
Reply
#17
(02/20/2016, 03:03 PM)marraco Wrote: I sampled 1000 points around the point r=1.1, with a radius R=.01.
...
I will attempt to implement a conjugate gradient on the red blue equation, use knesser.gp to get a very good first guess, and increase precision.
That also may lead nowhere, since the red blue eq is mined with local minima.


For real bases greater than eta=exp(1/e), the fixed points are complex conjugates of each other. For real bases between 1 and eta, you have an attracting fixed point, and a repelling fixed point. The branch at b=1 is pretty bad, worst than the branch at b=eta. For real bases less than b=1, you still have a real attracting fixed point, but the sexp(z) function is again not real valued. This is true when using either the Schroeder function or the more complicated Kneser solution.

So your basic problem is the half iterate isn't analyatic at b=1, no matter which approach you use, Kneser or the simpler Schroeder function solution. For simplicity, I'm graphing the Schroeder function half iterate, sexp(-0.5), as you loop around from 1.1 to -0.9 and then back to 1.1, but the Kneser solution you get using both fixed points (which is what fatou.gp gives you) isn't visually different for this graph. If it isn't analytic, increasing the precision of the number of sample points or decreasing the radius doesn't help.

Red is imaginary; magenta is real
   
- Sheldon
Reply
#18
(02/20/2016, 07:45 PM)sheldonison Wrote: So your basic problem is the half iterate isn't analytic at b=1, no matter which approach you use, Kneser or the simpler Schroeder function solution. For simplicity, I'm graphing the Schroeder function half iterate as you loop around from 1.1 to -0.9 and then back to 1.1, but the Kneser solution you get using both fixed points (which is what fatou.gp gives you) isn't visually different for this graph. If it isn't analytic, increasing the precision of the number of sample points or decreasing the radius doesn't help.

I do not understand if you speak of "half iterate", because your code uses it internally, or you say that I'm looking for the Taylor series of (I'm looking for ).

Now, 1.5 has been choosen by me arbitrarily. The Taylor series exist around 1 for integer values (the thread starts with the explicit formulas of those Taylor series).
I could had chosen any exponent =m+1/n (m and n integer).

Of course (I understand), knesser (and fatou?) give the Taylor series for a constant base , and I don't need the Taylor series of around 1. I already know that , and I need high precission calculating

Because is not 1, then the Taylor series should exist there, no matter if there is a singularity at 1.

Do you mean that is not possible to get higher precision for , with (nonlinear) conjugate gradient using knesser as initial guess?
I have the result, but I do not yet know how to get it.
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#19
(02/20/2016, 11:48 PM)marraco Wrote: I do not understand if you speak of "half iterate", because your code uses it internally, or you say that I'm looking for the Taylor series of (I'm looking for ).

I updated my post#17; I graphed the sexp(-0.5) which is not real valued for b=0.9; the graph used the simpler Schroeder function from the attracting fixed point instead of the more complicated Kneser solution, but both look nearly identical, and the problem is the function is not real valued at the real axis. So if you start with b=1.1, and rotate around 180 degrees to b=0.9, you get 0.90455-0.2937i; that is using the Schroeder function inverse Abel solution which isn't part of fatou.gp. Sexp(-0.5)=-0.90415-0.2939i. This is a problem because the Schroeder function Abel/inverse Abel solution is real valued for bases>1. So there's a complicated singularity at b=1. I don't claim to understand that singularity at b=1 yet, but I understand it well enough to know that the Taylor series won't converge, because it is a different function rotating clockwise around the singularity at b=1 than rotating counter clockwise around the singularity at b=1. The difference is fatal to getting a converging Taylor series.

To better understand the nature of the singularity, you will have to generate the Schroeder function, both for b=1.1 and b=0.9, and generate sexp(-0.5) for both cases yourself, and see that it is not real valued. My code actually works with the Schroeder function and its inverse for , which perhaps muddies the water; , even though there is always simple linear transform using sexpforminvabel(z), connecting the two. The fact that fatou.gp works with the bipolar Kneser tetration using both fixed points muddies the water even more, since Kneser tetration is no longer real valued for real bases<eta either; it has an even more complicated singularity at b=1, than the very complicated Schroeder function singularity at b=1. That is why I graphed the Schroeder function solution, even though both look visually identical.

So yes, the problem is you are looking for an analytic solution, where I don't think there is any simple form for an analytic solution for the sexp(-0.5), rotating around in a circle around b=1. Sorry.
- Sheldon
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#20
(02/21/2016, 01:38 PM)sheldonison Wrote: I updated my post#17; I graphed the sexp(-0.5) which is not real valued for b=0.9; the graph used the simpler Schroeder function from the attracting fixed point instead of the more complicated Kneser solution, but both look nearly identical, and the problem is the function is not real valued at the real axis.

(02/21/2016, 01:38 PM)sheldonison Wrote: So if you start with b=1.1, and rotate around 180 degrees to b=0.9,
You mean a circle centered at 1, with radius .1?

I don't need that. I need center at 1.1, with radius .01:

[Image: 0DhETMJ.png?1]

(02/21/2016, 01:38 PM)sheldonison Wrote: the Taylor series won't converge, because it is a different function rotating clockwise around the singularity at b=1 than rotating counter clockwise around the singularity at b=1. The difference is fatal to getting a converging Taylor series.
Ok. The taylor series for won't converge on 1, but is a different function, and we know that it exists and converge for
I have the result, but I do not yet know how to get it.
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