sexp for base (1/e)^e ~= 0.0660?

[sheldonison]

Here is the merged tetration solution for \( e^{-e} \). The graph extends from -4.12 to 6.12 in the real direction, and 2.12 to -6.12 in the imaginary direction, with grid lines every 2 units. At the origin, sexp(0)=1. You can see the branch singularity at sexp(-2), where from below the function quickly converges to the regular iteration from the fixed point of -0.1957+1.691i, and from above, as \( \Im(z) \) increases, the merged tetration function converges to \( \exp(-1) \) and to the novel pseudo 2-periodic solution I came up with earlier in this post. Near the origin, at z=0, you can see looking at the graph, that the function misbehaves as imag(z) decreases, before it starts behaving again. On the unit circle centered where sexp(0)=1, the function has a maximum magnitude near x=0.6-0.8i, of around 35000! This behavior makes it difficult to find an initial seed for the sexp(z) solution, or for Kouznetsov's method. I used a 10-term version of the "multi-term" exponential series that I posted earlier, as a starting seed for the upper half of the complex plane. The convergence was good enough to generate the lower half plane theta mapping from the other fixed point, using the path that is in my previous post. From there, I generated an initial sexp(z) taylor series on a unit circle, and then I iterated generating an upper half plane theta mapping and a lower half plane theta mapping, and an sexp Taylor series. I used the regular superfunction from the other fixed point, along with the 10-term "multi-term" pseudo 2-cyclic series I generated, from the exp(-e) fixed point. The sexp Taylor series, posted below the picture, appears to be accurate to more than 10 decimal digits.
   

I realize of course that this "multi-term" series I used for the upper half plane theta mapping doesn't have any of the strong theoretical underpinnings of a Schroder series, but it does seem to converge, and I think the results are at least interesting. This merged tetration solution has other singularities in the upper half of the complex plane, for real(z)<-2. For example, the second singularity is near z=-4.006+0.076i, and is visible near the left edge of the graph. Perhaps Mike can round off this sexp(z) as a starting seed for his new method. Also, Mike reports getting results for b=0.04, where both fixed points are repelling. Was that tetration solution as poorly behaving as the results for b=exp(-e)?

Code:

taylor series, for sexp(z=0), sexp(0)=1, for base exp(-e).  sexp(z-1) is a little better behaved; posted below.
{sexp=
        1.000000000000000000 +
+x^ 1* ( 0.2139536657103389784 + 2.528819288258810236*I)
+x^ 2* (-5.718005651873086168 + 1.405529567012070011*I)
+x^ 3* (-4.994351362455816907 - 9.591927497023705884*I)
+x^ 4* ( 12.47110195303975205 - 12.56438589624159071*I)
+x^ 5* ( 24.64512057624859248 + 11.06869551290638396*I)
+x^ 6* (-1.004557358147892928 + 39.11086617384984950*I)
+x^ 7* (-50.04117943660604027 + 21.13926212969639756*I)
+x^ 8* (-54.67981005276769267 - 48.16965142184401020*I)
+x^ 9* ( 23.82775802285444627 - 91.92324075543739587*I)
+x^10* ( 117.4806865647258164 - 27.71925893384273284*I)
+x^11* ( 100.3038670585419754 + 111.7102325184134650*I)
+x^12* (-58.72453764469395163 + 173.5093912929210437*I)
+x^13* (-215.3969244783325557 + 43.18987024699570534*I)
+x^14* (-173.4573393798765433 - 192.8890001580594647*I)
+x^15* ( 87.68637672585892981 - 289.0206055188629291*I)
+x^16* ( 335.6745790101675781 - 88.48727597802993670*I)
+x^17* ( 287.6967374498740693 + 269.6080191033535142*I)
+x^18* (-81.76939211911840563 + 435.3167743926416319*I)
+x^19* (-455.8704451399854484 + 186.4573068027563207*I)
+x^20* (-446.9925635830849074 - 307.4002340639193942*I)
+x^21* ( 9.792335057385973017 - 592.1415292176168827*I)
+x^22* ( 538.1068862554336423 - 348.5420477328105918*I)
+x^23* ( 634.3551614342157179 + 267.9271758681511601*I)
+x^24* ( 146.9682890290485080 + 719.2787656536446089*I)
+x^25* (-537.3989506388788575 + 561.4351267601670561*I)
+x^26* (-807.6471810000071224 - 124.9749933773521565*I)
+x^27* (-379.5389515922959864 - 764.9844982306551802*I)
+x^28* ( 419.2053192389169669 - 781.6054438260950562*I)
+x^29* ( 908.2091103456416840 - 118.1287493664723799*I)
+x^30* ( 644.1886143870421471 + 685.2831245521423183*I)
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+x^50* ( 625.0894076567540490 + 205.2906310615999909*I)
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+x^52* (-301.5819557894045742 + 507.6263889651549054*I)
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+x^54* (-384.2966259160203577 - 356.8098738534939122*I)
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+x^58* ( 162.4024069677279051 + 367.3931151200368648*I)
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+x^63* ( 204.7804045182772477 - 181.5132939397678168*I)
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+x^65* ( 119.6597353770539544 + 197.9520764881047600*I)
+x^66* (-68.83317111661640087 + 200.5526998240842019*I)
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+x^77* ( 31.44392156377771926 - 65.10469122467560931*I)
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+x^94* ( 6.367267327403528802 + 7.072760309264462004*I)
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+x^99* ( 1.691689330784595433 - 4.579332182908825997*I)
+x^100* ( 3.963565982411397253 - 1.552741910817760661*I)
+x^101* ( 3.306567482035113398 + 1.677478587920064020*I)
+x^102* ( 0.8265967777393779202 + 3.118163425612735045*I)
+x^103* (-1.541982463076915922 + 2.341395001833462984*I)
+x^104* (-2.407414692880514451 + 0.3578643678951269825*I)
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+x^106* (-0.07061481282813064643 - 1.827108924144911153*I)
+x^107* ( 1.135410688625148661 - 1.102115373557423992*I)
+x^108* ( 1.364796666435959455 + 0.09261603576750782346*I)
+x^109* ( 0.7293372577442321870 + 0.9293602239226706487*I)
+x^110* (-0.1740915282936671286 + 1.004249296614449025*I)
+x^111* (-0.7430084991260557095 + 0.4686487628569402322*I)
+x^112* (-0.7283568221024154604 - 0.2040165091003503814*I)
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+x^114* ( 0.2033364233832682973 - 0.5208695013047471519*I)
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+x^116* ( 0.3673233226645886248 + 0.1860976683124691950*I)
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+x^118* (-0.1613543579743463085 + 0.2554187841405899222*I)
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+x^120* (-0.1750561918587142027 - 0.1346540491596286680*I)
+x^121* (-0.01589846670949576749 - 0.1879051831132580688*I)
+x^122* ( 0.1091587112390239364 - 0.1181737147826808330*I)
+x^123* ( 0.1371209652284321828 + 0.0006316067093252268198*I)
+x^124* ( 0.07848840519920069173 + 0.08646528473247925902*I)
+x^125* (-0.009080516956819814497 + 0.09894814991054157720*I)
+x^126* (-0.06718865661864938160 + 0.05120510607926928580*I)
+x^127* (-0.07064641024685741826 - 0.01258996622771853822*I)
+x^128* (-0.03273236346456904970 - 0.05136317311892518464*I)
+x^129* ( 0.01323807564305556278 - 0.04992704326958708416*I)
+x^130* ( 0.03870987730627654353 - 0.02042646353229350574*I)
+x^131* ( 0.03493668581928837294 + 0.01235954274225916883*I)
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+x^133* (-0.01078120248096575492 + 0.02421145308742809885*I)
+x^134* (-0.02119552784591330546 + 0.007206257779886212512*I)
+x^135* (-0.01661899931891697422 - 0.008991449970517684863*I)
+x^136* (-0.003968390636353586240 - 0.01543414260234269119*I)
+x^137* ( 0.007259393508734014864 - 0.01129909148357653945*I)
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+x^140* ( 0.0008391333363945594403 + 0.007958289674598544314*I)
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+x^199* ( 0.00000005049524781613495621 - 0.00000008091701259963635920*I)
}

{sexp(-1)=
        4.5086319932657 E-12 - 6.0092568905711 E-12*I
+x^ 1* (-0.078709154986661 - 0.93030062659886*I)
+x^ 2* ( 0.93567569573982 - 0.31802420239162*I)
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}