 Expansion of base-e pentation - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Computation (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=8) +--- Thread: Expansion of base-e pentation (/showthread.php?tid=372) Pages: 1 2 Expansion of base-e pentation - andydude - 10/24/2009 So I have been revamping the functions I use for tetration, and I recently solved all the mumbo-jumbo to use tetration as the function in the NaturalIterate function. So now you can do: Code:```< 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). [attachment=791] 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 .... [attachment=792] 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```