10/30/2010, 11:11 PM
(This post was last modified: 10/30/2010, 11:12 PM by sheldonison.)

(10/24/2009, 04:30 AM)andydude Wrote: .....

I'm still working on doing this for other bases, but this is enough to get an expansion of pentation about zero.

What follows below is InverseSeries[...] of the output from above. In other words, the above gives the coefficients of the base-e penta-logarithm about (0), and the following gives the coefficients of the base-e penta-exponential about (-1).

I read through the pentation posts scattered through this forum, and decided to generate some results of my own. First off go to my fast accurate kneser.gp post, and download the latest version of the pari-gp kneser.gp code. Then, download the pentation.gp code. After the downloads, start gp.

pentation.gp (Size: 3.67 KB / Downloads: 387)

Code:

`\r kneser.gp`

\r pentation.gp

gp > init;loop; /* 13 sexp(z) loops .... */

gp > genpent /* generate pentation base e */

complex sexp Taylor series centered at -1.8503545290271814184834459502910

pentation base 2.71828182845904523536029

pentation(-0.5) 0.491054338635648197413514

sexp fixed point -1.85035452902718141848345

sexp slope at fixed 6.46067129568183939020883

pentation period 3.36767615671259898023746*I

pentation singularity -2.31527062760141112146561 + 1.68383807835629949011873*I

pentation precision, via sexp(pent(-0.5))-pent(0.5)

-9.69394178147793704836998 E-21

gp > ploth(t=-10,1.5,pent(t));

So what's all this telling you? Its telling you that the fixed point for sexp(z) to generate pentation base e, pfixed, is -1.85035.... Its telling you the slope of sexp(pfixed)=-6.46067... which leads to the pentation complex periodic period, pperiod=3.367*I. At imag(z)=0, the function is real valued for all z. Here is the graph, generated by that last line ploth line. Notice, it starts out at the real valued fixed point, pfixed, and then grows somewhat intermittantly, with pent(-inf)=-1.85..., pent(-2)=-1, pent(-1)=0, and pent(0)=1, and pent(2)=e, and pent(3)=sexp(e).

At imag(z)=imag(pperiod/2), there is another real valued line, starting at the fixed point and growing towards -infinity, which is the first singularity that occurs at -2.315 + 1.6838*I. Then there is a slew of singularities after that, corresponding to sexp(z)=-2,-3,-4,-5 ....

Continuing, to generate the Taylor series, centered at pentation(z=-1). Notice that the first term of the Taylor series is almost zero, but not quite due to precision errors. The pentation taylor series is accurate to approximately 20 decimal digits. More accurate results are possible by setting gp's precision to "\p 134". Then type in the sexp initialization command, "init;loop". Later, I will post pentation plots for other bases. So far, I've only had time to try bases between B=1.6 and B=e, all of which worked fine. I'll post more later. Here are the Taylor series terms, followed by Andy's terms for historical comparison.

- Sheldon

Code:

`gp > pentaylor(-1,1)`

gp > for (s=1,30,print(real(ptseries[s])))

6.7242536178550628294755514087 E-24

0.99727185142263340743455208346122

-0.045007215859218115832617467992327

0.0088901369292365764437286761921372

0.045713734782598722205971510001068

-0.010706554884752458976051797391420

0.00011329335331439235574805971805731

0.0051620130076806122858704585184006

-0.0012422756898373028878826856222621

-0.00067376885079665208568672130450693

0.00050296665968765950574361816768155

0.000039905534193068199638492988158461

-0.000094623078715532686231662582532929

0.000026746817775170179559855402990613

0.000015560915176630839373361742908680

-0.000014806164180600879049897255662325

-0.0000010862859329576915398646271649914

0.0000059140073162222162194013397871868

-0.00000071091367653831526613315587588080

-0.0000017666311876111783264226773258896

0.00000051445590441872869647430881209296

0.00000036270009896115685098739030842790

-0.00000020971694575358607315821487792583

-0.000000021225391058732913781072384245741

0.000000064546351710396107513893801263449

-0.000000022236468044317568271338433775775

-0.000000015692632671055194971436603480275

0.000000013400661392837577306043949458764

0.0000000026987924084642996469868738785694

-0.0000000049614802252464409094532195966523

Andy's results are pretty reasonable, all considering, but only for the first few terms. Here are Andy's coefficients for historical record.

Code:

`0,`

0.997386001614238200000,

-0.044854069033065140000,

0.008127184531878105000,

0.045268576293608810000,

-0.009169795166599723000,

0.000529626080101428000,

0.003682350459440369500,

-0.001300714479652927000,

0.000136554270543782140,

0.000349632018705509600,

-0.000212903018660854500,

0.000030850789704285015,

0.000053653522961255240,

-0.000028243223065159680,

-0.000003800898968414997,

0.000000972449120890964

0.000005775482651540000,

0.000010790317715530437,

-0.000029357772002764790,

0.000020775705975594905