# Tetration Forum

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$\vspace{15}{e^{i.\theta}}$ is an interesting and important function, so what we get if we do the same with tetration: $\vspace{15}{^{i.\theta}a}$?

I put i.x in the polynomials obtained by minimizing the error $\vspace{15}{\Delta^2 \,=\, (^{x}a-a^{^{x-1}a})^2}$, and drawed the imaginary vs real part:

These functions probably are periodical, but my polynomials only converge on -1≤x≤1, so, be careful about looking at values outside of that range.
For example, this chart is only valid in the range 0..4, for real and -4..3 for the imaginary axis.

This is base $\vspace{15}{e^{-e}}$

This is base $\vspace{15}{e^{\frac{1}{e}}}$

This is a zoom in the curl transition

base e, or something close, seems to delimit the transition towards the negative axis:

Many of these seem to be described by something like $\vspace{15}{^{i.x}a\,=\,c+a.cos(n.x) \,+\, i. b.sin(m.x)}$
(04/18/2015, 11:20 PM)marraco Wrote: [ -> ]$\vspace{15}{e^{i.\theta}}$ is an interesting and important function, so what we get if we do the same with tetration: $\vspace{15}{^{i.\theta}a}$?

Your graph for base(e), and the other bases<1 is not correct. The general form for the unique slog for bipolar tetration, which is real valued and analytic, from both fixed points, for bases $b>\exp(1/e)$ is:

$\text{slog}(z) = \alpha(z) +\frac{\ln(z-L)}{\ln(L\cdot \ln(b))} + \frac{\ln(z-L^{*})}{\ln(L^{*}\cdot \ln(b))}\;\;b^L=L\;\;$ This is also the form for Jay's accelerated slog

Here; $\alpha(z)$ is analytic and bounded on a sickle including the fixed points. slog(z) has a singularity at the fixed points themselves, but is bounded between the fixed points, and is analytic, and has an inverse, sexp(z), on that sickle. Here, at L, $\alpha(L)$ is a constant. off topic: actually, $\alpha(z)$ has a complex singularity at z=L, but it is continuous.

In the upper half of the complex plane, as $\Im(z)$ gets arbitrarily large, the sexp(z) approaches the Koenig's solution; where the exponential term below goes to zero, and the sexp(z) approaches L. In the lower half of the complex plane, it approaches the conjugate. This is the basis for Kouznetsov's solution.
$\text{sexp}(z) \approx L + (L\cdot\ln(b))^{(z+k)}$

So then here is a graph of sexp(z) base e, from -5i to +5i, showing the sexp(z) function going from L* to L. You can generate sexp(z) for any complex(z) for many real bases $b>\eta$ using my implementation of Kneser's solution
[attachment=1194]
(04/19/2015, 02:40 PM)sheldonison Wrote: [ -> ]So then here is a graph of sexp(z) base e, from -5i to +5i, showing the sexp(z) function going from L* to L. You can generate sexp(z) for any complex(z) for many real bases $b>\eta$ using my implementation of Kneser's solution

I still need a crash course in PariGP.

Do you think that $\vspace{15}{^xa=a^{^{x-1}a}}$ is still valid on the complex plane?

Did you checked its accuracy on the real line?

I wish I could generate the points and verify.
(04/19/2015, 08:40 PM)marraco Wrote: [ -> ]Do you think that $^xa=a^{^{x-1}a}$ is still valid on the complex plane?

Did you checked its accuracy on the real line?
Yes to both. The pari-gp algorithm I wrote will generate a ~33 decimal digits accurate solution in a few seconds, for base(e). The algorithm generates a Taylor series at the real axis, as well as a 1-cyclic $\theta(z)$ mapping for the complex valued Koenig's solution. Both solutions are equal, but the Koenig's theta mapping can be used anywhere in the upper half of the complex plane, except right near the real axis, where an arbitrarily large number of terms in the $\theta(z)$ mapping would be required due to the $\theta(z)$ singularity for integer values of z. $\text{sexp}(z) = \text{Koenig}(z+\theta(z))$. The 109 term Taylor series approximation for $\text{sexp}(z)=\sum_{n=0}^{\infty}a_n\cdot z^n\;\;$ would be accurate to ~33 decimal digits within a radius<=1; the radius of convergence of the infinite series is 2.

By definition, $\text{Koenig}(z+1) = \exp(\text{Koenig}(z))\;$ and is entire for repelling fixed points, so if $\theta(z)=\sum_{n=0}^{\infty}a_n\cdot \exp(z \cdot 2\pi i n)$
then $\theta(z+1)=\theta(z)$ and therefore $\text{Koenig}(z+1+\theta(z+1)) = \exp(\text{Koenig}(z+\theta(z)))\;\;\;\$ and if $\text{sexp}(z)=\text{Koenig}(z+\theta(z))\;$, then $\text{sexp}(z+1)=\exp(\text{sexp}(z))$.

For ~33 decimal digits of accuracy, the 1-cyclic approximation would require 94 terms if $\Im(z)>=0.12i$. Also, $\theta(z)$ goes to a constant as $\lim_{z \to i \infty}$. There is also a complex conjugate version of the $\theta(z)$ mapping for the Im(z)<0 half of the complex plane.

More iterations requires more precision, so it runs a lot slower. Generating such a >70 digit accuracy solution requires 28 iterations and takes 40 seconds. My $\theta(z)$ mapping is mathematically equivalent to Kneser's Riemann mapping.
Code:
{sexp= 1 +x^ 1*  1.0917673512583209918013845500271517 +x^ 2*  0.27148321290169459533170668362354901 +x^ 3*  0.21245324817625628430896763774094856 +x^ 4*  0.069540376139987373728674232707469711 +x^ 5*  0.044291952090473304406440344385514804 +x^ 6*  0.014736742096389391152096286915534111 +x^ 7*  0.0086687818172252603663803925296399219 +x^ 8*  0.0027964793983854596948259913011495569 +x^ 9*  0.0016106312905842720721626451640260303 +x^10*  0.00048992723148437733469866722583243492 +x^11*  0.00028818107115404581134526404129643988 +x^12*  0.000080094612538543333444273583009977844 +x^13*  0.000050291141793805403694590114624236685 +x^14*  0.000012183790344900091616191711098613934 +x^15*  0.0000086655336673815746852458045541327926 +x^16*  0.0000016877823193175389917890093176351289 +x^17*  0.0000014932532485734925810665044317554713 +x^18*  0.00000019876076420492745531981897955236756 +x^19*  0.00000026086735600432637316458216085893667 +x^20*  0.000000014709954142541901861412188201900914 +x^21*  0.000000046834497327413506255093709947352846 +x^22* -0.0000000015492416655467695218054651322677986 +x^23*  0.0000000087415107813509359129925581627587247 +x^24* -0.0000000011257873101030623175751344777492435 +x^25*  0.0000000017079592672707284125656087948039197 +x^26* -0.00000000037785831549229851764921435415800623 +x^27*  0.00000000034957787651102163178731457999981781 +x^28* -1.0537701234450015066294258497294163 E-10 +x^29*  7.4590971476075052807322860078252226 E-11 +x^30* -2.7175982065777348693298760969650549 E-11 +x^31*  1.6460766106614471303885109354654427 E-11 +x^32* -6.7418731524050529991474333392275879 E-12 +x^33*  3.7253287233194685443170989538379133 E-12 +x^34* -1.6390873267935902234582225010471847 E-12 +x^35*  8.5836383113585680604885670142883911 E-13 +x^36* -3.9437387391053843135799065799345925 E-13 +x^37*  2.0025231280218870558933488201935127 E-13 +x^38* -9.4419622429240650237203369464349681 E-14 +x^39*  4.7120547458493713408134162135725801 E-14 +x^40* -2.2562918820355970800481578125058583 E-14 +x^41*  1.1154688506165369962907704017701283 E-14 +x^42* -5.3907455570163504918533616427279775 E-15 +x^43*  2.6521584915166818728179847001994855 E-15 +x^44* -1.2889107655445536819394124602816863 E-15 +x^45*  6.3266785019566604528387812745585687 E-16 +x^46* -3.0854571504923359890162184894469925 E-16 +x^47*  1.5131767717827405270960446817215775 E-16 +x^48* -7.3965341370947514333418316897203391 E-17 +x^49*  3.6269876710541876035237240096591600 E-17 +x^50* -1.7757255986762984029994538367761954 E-17 +x^51*  8.7098795443960546502631507727982294 E-18 +x^52* -4.2692892823391563141718062049000602 E-18 +x^53*  2.0950441625755281093483690055067076 E-18 +x^54* -1.0278837092822587892363235029846949 E-18 +x^55*  5.0468242474381763889822749445669773 E-19 +x^56* -2.4780505958215521454269990958399214 E-19 +x^57*  1.2173942030393317020113788775364947 E-19 +x^58* -5.9816486323037815150562410082210709 E-20 +x^59*  2.9402643445138969080986317689280122 E-20 +x^60* -1.4455835436201850220023758753334795 E-20 +x^61*  7.1095903736074123276947876938966522 E-21 +x^62* -3.4974025487872032482054204148733769 E-21 +x^63*  1.7209641671884893968788892238552706 E-21 +x^64* -8.4703003156215453164497615979164144 E-22 +x^65*  4.1700185394242509196286233654876760 E-22 +x^66* -2.0534096527658659704995748057858764 E-22 +x^67*  1.0113838615494863919451981634503253 E-22 +x^68* -4.9825424396849504025363950795013587 E-23 +x^69*  2.4551694611351628954393221798701698 E-23 +x^70* -1.2100465472324105299447061203120237 E-23 +x^71*  5.9650225602500368204601428440073432 E-24 +x^72* -2.9410859968904307308168910746569497 E-24 +x^73*  1.4503990988357537983984682685714610 E-24 +x^74* -7.1539937380082747218197703756972717 E-25 +x^75*  3.5293042057319186187079609197431644 E-25 +x^76* -1.7414327791202733830945057246997563 E-25 +x^77*  8.5940845951550628378204439831675622 E-26 +x^78* -4.2419517528160810175697295824342703 E-26 +x^79*  2.0941281695727407230153052186977015 E-26 +x^80* -1.0339757529559312470973226618639183 E-26 +x^81*  5.1060532208314709608618429841692590 E-27 +x^82* -2.5218920904122435678668059429207785 E-27 +x^83*  1.2457539467832514631761121316659649 E-27 +x^84* -6.1546176441900236712036669408929694 E-28 +x^85*  3.0411051132916308845009731999557020 E-28 +x^86* -1.5028717768397369996350257200319465 E-28 +x^87*  7.4279860680522329081644760189512429 E-29 +x^88* -3.6717893955394307351304683981660285 E-29 +x^89*  1.8152677927719233922578013095639624 E-29 +x^90* -8.9754810772507114316889719589853523 E-30 +x^91*  4.4384444557718439080961665330633816 E-30 +x^92* -2.1951051980324132441203842782335479 E-30 +x^93*  1.0857264522839554703474518850312616 E-30 +x^94* -5.3713051049816620023559234957667223 E-31 +x^95*  2.6569463100450525421735771247391265 E-31 +x^96* -1.3150936158893411039625390247367046 E-31 +x^97*  6.5031082874542227062257294778935637 E-32 +x^98* -3.2232822252628017181294513063989031 E-32 +x^99*  1.5906201065416840874596884648767165 E-32 +x^100* -7.8953030215835351071271084092673743 E-33 +x^101*  3.8940334244238877083713174341717584 E-33 +x^102* -1.9230987131520699822802998663703356 E-33 +x^103*  9.5400148793742290635690623957050238 E-34 +x^104* -4.7203747902964989216918644128748847 E-34 +x^105*  2.3432959042953020194125107406188183 E-34 +x^106* -1.1829537815826610344143523964787138 E-34 +x^107*  4.9057060464151121689787019274724467 E-35 +x^108* -1.1579374110263163325549928790619021 E-35 }

(04/19/2015, 11:19 PM)sheldonison Wrote: [ -> ]
Code:
{sexp= 1 +x^ 1*  1.0917673512583209918013845500271517 +x^ 2*  0.27148321290169459533170668362354901 +x^ 3*  0.21245324817625628430896763774094856 +x^ 4*  0.069540376139987373728674232707469711 +x^ 5*  0.044291952090473304406440344385514804 +x^ 6*  0.014736742096389391152096286915534111 +x^ 7*  0.0086687818172252603663803925296399219 +x^ 8*  0.0027964793983854596948259913011495569 +x^ 9*  0.0016106312905842720721626451640260303 +x^10*  0.00048992723148437733469866722583243492 +x^11*  0.00028818107115404581134526404129643988 +x^12*  0.000080094612538543333444273583009977844 +x^13*  0.000050291141793805403694590114624236685 +x^14*  0.000012183790344900091616191711098613934 +x^15*  0.0000086655336673815746852458045541327926 +x^16*  0.0000016877823193175389917890093176351289 +x^17*  0.0000014932532485734925810665044317554713 +x^18*  0.00000019876076420492745531981897955236756 +x^19*  0.00000026086735600432637316458216085893667 +x^20*  0.000000014709954142541901861412188201900914 +x^21*  0.000000046834497327413506255093709947352846 +x^22* -0.0000000015492416655467695218054651322677986 +x^23*  0.0000000087415107813509359129925581627587247 +x^24* -0.0000000011257873101030623175751344777492435 +x^25*  0.0000000017079592672707284125656087948039197 +x^26* -0.00000000037785831549229851764921435415800623 +x^27*  0.00000000034957787651102163178731457999981781 +x^28* -1.0537701234450015066294258497294163 E-10 +x^29*  7.4590971476075052807322860078252226 E-11 +x^30* -2.7175982065777348693298760969650549 E-11 +x^31*  1.6460766106614471303885109354654427 E-11 +x^32* -6.7418731524050529991474333392275879 E-12 +x^33*  3.7253287233194685443170989538379133 E-12 +x^34* -1.6390873267935902234582225010471847 E-12 +x^35*  8.5836383113585680604885670142883911 E-13 +x^36* -3.9437387391053843135799065799345925 E-13 +x^37*  2.0025231280218870558933488201935127 E-13 +x^38* -9.4419622429240650237203369464349681 E-14 +x^39*  4.7120547458493713408134162135725801 E-14 +x^40* -2.2562918820355970800481578125058583 E-14 +x^41*  1.1154688506165369962907704017701283 E-14 +x^42* -5.3907455570163504918533616427279775 E-15 +x^43*  2.6521584915166818728179847001994855 E-15 +x^44* -1.2889107655445536819394124602816863 E-15 +x^45*  6.3266785019566604528387812745585687 E-16 +x^46* -3.0854571504923359890162184894469925 E-16 +x^47*  1.5131767717827405270960446817215775 E-16 +x^48* -7.3965341370947514333418316897203391 E-17 +x^49*  3.6269876710541876035237240096591600 E-17 +x^50* -1.7757255986762984029994538367761954 E-17 +x^51*  8.7098795443960546502631507727982294 E-18 +x^52* -4.2692892823391563141718062049000602 E-18 +x^53*  2.0950441625755281093483690055067076 E-18 +x^54* -1.0278837092822587892363235029846949 E-18 +x^55*  5.0468242474381763889822749445669773 E-19 +x^56* -2.4780505958215521454269990958399214 E-19 +x^57*  1.2173942030393317020113788775364947 E-19 +x^58* -5.9816486323037815150562410082210709 E-20 +x^59*  2.9402643445138969080986317689280122 E-20 +x^60* -1.4455835436201850220023758753334795 E-20 +x^61*  7.1095903736074123276947876938966522 E-21 +x^62* -3.4974025487872032482054204148733769 E-21 +x^63*  1.7209641671884893968788892238552706 E-21 +x^64* -8.4703003156215453164497615979164144 E-22 +x^65*  4.1700185394242509196286233654876760 E-22 +x^66* -2.0534096527658659704995748057858764 E-22 +x^67*  1.0113838615494863919451981634503253 E-22 +x^68* -4.9825424396849504025363950795013587 E-23 +x^69*  2.4551694611351628954393221798701698 E-23 +x^70* -1.2100465472324105299447061203120237 E-23 +x^71*  5.9650225602500368204601428440073432 E-24 +x^72* -2.9410859968904307308168910746569497 E-24 +x^73*  1.4503990988357537983984682685714610 E-24 +x^74* -7.1539937380082747218197703756972717 E-25 +x^75*  3.5293042057319186187079609197431644 E-25 +x^76* -1.7414327791202733830945057246997563 E-25 +x^77*  8.5940845951550628378204439831675622 E-26 +x^78* -4.2419517528160810175697295824342703 E-26 +x^79*  2.0941281695727407230153052186977015 E-26 +x^80* -1.0339757529559312470973226618639183 E-26 +x^81*  5.1060532208314709608618429841692590 E-27 +x^82* -2.5218920904122435678668059429207785 E-27 +x^83*  1.2457539467832514631761121316659649 E-27 +x^84* -6.1546176441900236712036669408929694 E-28 +x^85*  3.0411051132916308845009731999557020 E-28 +x^86* -1.5028717768397369996350257200319465 E-28 +x^87*  7.4279860680522329081644760189512429 E-29 +x^88* -3.6717893955394307351304683981660285 E-29 +x^89*  1.8152677927719233922578013095639624 E-29 +x^90* -8.9754810772507114316889719589853523 E-30 +x^91*  4.4384444557718439080961665330633816 E-30 +x^92* -2.1951051980324132441203842782335479 E-30 +x^93*  1.0857264522839554703474518850312616 E-30 +x^94* -5.3713051049816620023559234957667223 E-31 +x^95*  2.6569463100450525421735771247391265 E-31 +x^96* -1.3150936158893411039625390247367046 E-31 +x^97*  6.5031082874542227062257294778935637 E-32 +x^98* -3.2232822252628017181294513063989031 E-32 +x^99*  1.5906201065416840874596884648767165 E-32 +x^100* -7.8953030215835351071271084092673743 E-33 +x^101*  3.8940334244238877083713174341717584 E-33 +x^102* -1.9230987131520699822802998663703356 E-33 +x^103*  9.5400148793742290635690623957050238 E-34 +x^104* -4.7203747902964989216918644128748847 E-34 +x^105*  2.3432959042953020194125107406188183 E-34 +x^106* -1.1829537815826610344143523964787138 E-34 +x^107*  4.9057060464151121689787019274724467 E-35 +x^108* -1.1579374110263163325549928790619021 E-35 }
Well, I compared your coefficients against the ones I got with excel, and it looks like Excel can barely work with 7 coefficients:

This is what I get when I graph with these coefficients, and compare the polynomial with ln(x+1) and exp(x-1).
Excel seems to be capable to ccalculate the polynomial (with your coefficients) just between -1.9≤x≤1.3

(04/19/2015, 02:40 PM)sheldonison Wrote: [ -> ]

I wonder what it does beyond this range. There is loop or not?
(04/18/2015, 11:20 PM)marraco Wrote: [ -> ]
Numerically, when I try to minimize the error $\vspace{15}{\Delta^2=\sum_{}^{}{(^xa \,-\, a^{^{x-1}a})}}$, I get similar coefficients to the ones posted by Sheldonison.

But I can also make it converge to different values of the first derivative at zero, or force some restriction.

Does that means that there are more than one solution to tetration? (a finite or infinite number of solutions)
There is needed an additional restriction when defining what tetration is?

Here are the derivatives I got for each base:

For bases a>1, I do ever converge to the same values. The same goes for bases between 0.1 and 0.6.

The other derivatives I get are highly variable, even by sign.
I wonder if in that range there are more than a single "correct" derivative at zero, or if I just get artifacts due to local minima when minimizing the error.

These are the derivatives that I get for bases a>1
Code:
2.718281828    1.09562140557617 2.7    1.09143052409660 2.6    1.06822431122401 2.5    1.04307153925875 2.4    1.01659572039055 2.3    0.98764740646929 2.2    0.95793010260192 2.1    0.92527675767840 2    0.89028921820469 1.9    0.85186626289898 1.8    0.80938670730807 1.7    0.76259264471247 1.6    0.70969066859471 1.5    0.64958552115556 1.444667861    0.61224627377907 1.42    0.59451852096765 1.4    0.57955308252839 1.35    0.53981702850685 1.3    0.49599575015175 1.25    0.44716716706888 1.2    0.39190949073500 1.15    0.32798289733884 1.1    0.25143159620504 1.05    0.15333702806960 1.01    0.04148265011217 1    0.00000000000000

$\vspace{15}{\frac{\mathrm{d} \,^xa}{\mathrm{d} x}\mid_{x=0} \,=\, a_1 \,=\, \,-\,0.6452164561.a^6 \,+\, 7.6244443943.a^5 \,-\, 37.0680451104.a^4 \,+\, 94.9910696302.a^3 \,-\, 135.6878517026.a^2 \,+\, 103.3197783493.a^ \,-\, 32.5227109180}$

I wonder if there are some tool on Internet capable of proposing different equations matching it.

For example, this website:

Finds that the derivative got by Sheldonison for base e (a1=1.09176735125832) is nearly imaginary part of
$\vspace{15}{ i-((7- i\,.19)^{\frac{1}{16}}) \,=\, -1.2033454171730380657324537148862244072935639001352 + i \,.\, 1.0917673511359367268632631345613715697821843358013}$

Meanwhile, this place
http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html

Finds

$\vspace{15}{(atan(1/2)*Cahen +exp(-1/2*Pi))/atan(1/2) \,=\, 1.0917673852236835355691837676191405901326411536231}$

where the Cahen constant is

$\vspace{15}{C = \sum\frac{(-1)^i}{s_i-1}=\frac11 - \frac12 + \frac16 - \frac1{42} + \frac1{1806} - \cdots\approx 0.64341054629.}$
(04/19/2015, 08:40 PM)marraco Wrote: [ -> ]I still need a crash course in PariGP.

What do you need most?

Gottfried
I ran Shedonison program, and generated $\vspace{15}{^{i.x}2}$ for -i.100<x<i.100.
Then I graphed the imaginary vs real part:

There is no loop. The function converges very fast to the upper and lower point, where it do spirals too small to be seen.

(04/20/2015, 07:53 AM)Gottfried Wrote: [ -> ]What do you need most?
Well I can't run Sheldonison code for real values. I get this error:

Any idea ?

There is no problem if I run sexp(I.100)

(actually, I don't know the radius of convergence of the polynomial)
(04/19/2015, 02:40 PM)sheldonison Wrote: [ -> ]So then here is a graph of sexp(z) base e, from -5i to +5i, showing the sexp(z) function going from L* to L. You can generate sexp(z) for any complex(z) for many real bases $b>\eta$ using my implementation of Kneser's solution

I get crashes for almost all bases

Code:
(23:15) gp > init(0.5)   ***   at top-level: init(0.5)   ***                 ^---------   ***   in function init: rawinit(initbase);lo   ***                     ^--------------------   ***   in function rawinit: ...lastabs=curabs;lastl=L;L=log(L)*rlnB;curabs=a   ***                                                    ^--------------------   *** log: domain error in log: argument = 0
(04/21/2015, 03:20 AM)marraco Wrote: [ -> ]...
I get crashes for almost all bases

[code]
(23:15) gp > init(0.5)
The kneser.gp program will only generate useful results for you for real bases>exp(1/e). It will take a couple of weeks of work and experimenting to figure out how to rewrite the algorithms to work for bases between 0 and 1, which is actually a different function (in many ways) than tetration for real bases>eta. I haven't done that yet; and things are very busy now
Try "init(exp(1));"
or init(1.5); /* runs a little slower for bases closer to exp(1/e) */
or init(2);
or init(3);
or init(4);

As I remember, the program will generate results for bases between 1 and eta as well, but what it does for bases between 1 and eta is not what you might expect it to be; so I would recommend avoiding those bases as well. Kneser.gp accurately generates Kneser's merged tetration solution from both complex repelling fixed points for most real bases>eta, for much of the complex plane, unless the iterated function overflows. pari-gp's internal representation has limitations, and sexp(10) is a humongous number. The largest number pari-gp can represent is about $\text{sexp}_e\left(4.08\right)\approx 7.24676\cdot 10^{131475196{$
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