fixed point formula

[sheldonison]

\( z=i\sqrt{(\ln(\ln(b))+1)} \)
\( L(z)=\sum_{n=0}^{\infty} a_n\times z^n \)
....

This evening, I stumbled across this equivalent formal power series for the same fixed-point function.... Instead of the fixed point function above, the function is the ln(ln(fixed-point)) of the fixed-point function quoted above.
\( z=\sqrt{-2\left(\ln(\ln(b))+1\right)} \)
\( L(b)=\exp(\exp(\text{xfixed}(z))) \)

Here are the first 60 terms of the "xfixed" formal power series; update xfixed is defined as follows: see my answer to this mathstack question
\( g(x)=\sqrt{2(\exp(x)-x-1)};\;\;\;\;\text{xfixed}=g^{-1}(x) \)

Code:

a1=   1
a2=  -1/6
a3=   1/36
a4=  -1/270
a5=   1/4320
a6=   1/17010
a7=  -139/5443200
a8=   1/204120
a9=  -571/2351462400
a10= -281/1515591000
a11=  163879/2172751257600
a12= -5221/354648294000
a13=  5246819/10168475885568000
a14=  5459/7447614174000
a15= -534703531/1830325659402240000
a16=  91207079/1595278956070800000
a17= -4483131259/2987091476144455680000
a18= -2650986803/818378104464320400000
a19=  432261921612371/337123143997663268044800000
a20= -6171801683/24551343133929612000000
a21=  6232523202521089/1189370452023756009662054400000
a22=  4283933145517/279517041579788632620000000
a23= -25834629665134204969/4267461181861236562667451187200000
a24=  11963983648109/10062613496872390774320000000
a25= -1579029138854919086429/76814301273502258128014121369600000000
a26= -208697624924077/2747093484646162681389360000000
a27=  746590869962651602203151/24887833612614731633476575323750400000000
a28= -29320119130515566117/4968750398484053826709377112800000000
a29=  1511513601028097903631961/17321932194379853216899696425330278400000000
a30=  2700231121460756431181/6931406805885255088259581072356000000000
a31= -8849272268392873147705987190261/57510547078560550665428682101739057315840000000000
a32=  10084288256532215186381/332707526682492244236459891473088000000000
a33= -6208770108287283939483943525987/15842905490859115174615483904200811615354880000000000
a34= -6782242429223267933535073/3308776352857385368931593620699860160000000000
a35=  2355444393109967510921431436000087153/2907806873792281999148925915777016963882234675200000000000
a36= -51748587106835353426330148693/323935822497443742389140878653757709384320000000000
a37=  2346608607351903737647919577082115121863/1278155589444135475545901875538945576644075073830912000000000000
a38=  7007277101869903281324331583/636701444219113562626932071836696187410560000000000
a39= -2603072187220373277150999431416562396331667/598176815859855402555482077752226529869427134552866816000000000000
a40=  585302872633292617248814587726421/681334215458873423367080010072448590148040256000000000000
a41= -73239727426811935976967471475430268695630993/8240483815285368025604321103114672675481228205600293257216000000000000
a42= -110855495796575034381969281033555329/1845734389678088103901419747286263230711041053504000000000000
a43=  34856851734234401648335623107688675640839679447003/1466970928797101215918081242776474029689168245160964205649592320000000000000
a44= -18447986573777204063499607563765439/3929628055443671447015925913577205587965442242944000000000000
a45=  909773124599542506852275229422593983242880452145053/20596271840311301071489860648581695376835922162059937447320276172800000000000000
a46=  38650132745379700438031566826935471987259957/116354250486669482021689508823610789591151200820537602492800000000000000
a47= -1527335577854677023023224272800947125313629267269390501/11616297317935573804320281405800076192535460099401804720288635761459200000000000000
a48=  217784448556937372678947372805330071920344629/8377506035040202705561644635299976850562886459078707379481600000000000000
a49= -183856455668177802003316143799518064719008299958634826921/819645938753534087632839055993253376145302064613791341063566139328561152000000000000000
a50= -1167289109751840227800236733417523750884898531/628312952628015202917123347647498263792216484430903053461120000000000000000
a51=  2583312098861137963745902036370496943872138148651712093816393/3511363201620140031419082515875097463406474044805482105116317340883555975168000000000000000
a52= -107748081854646619391722638838074116233224341741059/740272037656801231924925556964805979417351539791645668557356972800000000000000000
a53=  5180134290822682443757710427952467581918233549140896702364013/4466453992460818119965072960193123973453034984992573237707955657603883200413696000000000000000
a54=  27346403208634415483181063970969158506217340077059/2607045002182647816779085657136925405774151074918404311006344121600000000000000000
a55= -527550309097873396592733540579928993424142983691519876840948418433873/126949575182314587341955272240273143385058268074945409878049452860509411817038440038400000000000000000
a56=  377036553764192941179202019520271416437725306277603527/458349795923735677963564607552557129439965048783554498726647372706739200000000000000000
a57= -2114866241537081164613223324215572812504648703648482437460602956015127/347334037698812710967589624849387320301519421453050641426343303026353750731417171945062400000000000000000
a58= -2107144283266473668026539971128155003797327940672559775477/35290642537148028524804656958509136181230108931089778629458214461555384704000000000000000000
a59=  180394412915538782140015777241228025103785450235726235175126981743099027459/7623287459413541380316657086194352905977748262051555478025382814822412121053144089850229555200000000000000000
a60= -907975882295290895046750344009772888231118193554130319911809/193745627528942676601177566702215157634953298031682884675725597393939062024960000000000000000000

The series above calculates the fixed point x for f(x) function, as z varies in the neighborhood of zero.  Here, if x is the desired fixed point L, then f(x)=x
\( f(x)=\exp(x)-1+k \;\mapsto \; f(x) = \exp(x)-1-\frac{ z^2}{2}\;\;\; k \mapsto -\frac{ z^2}{2};\; \) this mapping gives rational coefficients for L(z)
k=0 corresponds to the parabolic case, or \( f(x)=\exp(x)-1\; \) which has a parabolic fixed point of zero.
\( L(z) = \sum_{n=1}^{\infty}a_n z^n\;\; \) The Taylor series from above
\( L(z) = \exp(L(z))-1-\frac{ z^2}{2}\;\; \)plugging L(z) into f(x) as x
For \( \text{sexp}_e; \; k=1; \; f(x)=\exp(x); \; z=\pm\sqrt{-2} \)

With an 80 term formal power series, one gets the fixed point for base(e) accurate to 35 decimal digits.  Smaller bases closer to exp(1/e) are more accurate.  For base(2), the fixed point is accurate to 43 decimal digits.

So for example, for tetration base(e), the fixed point is \( L\approx 0.318132 + 1.337236i \)
To get this value for L and its conjugate, plug \( z=\pm\sqrt{-2(\ln(\ln(e))+1)}=\pm\sqrt{-2} \) into the series above...

\( L = \sqrt{-2}
- \frac{\left(\sqrt{-2}\right)^2}{6}
+ \frac{\left(\sqrt{-2}\right)^3}{36}
- \frac{\left(\sqrt{-2}\right)^4}{270}
+ \frac{\left(\sqrt{-2}\right)^5}{4320}
+ \frac{\left(\sqrt{-2}\right)^6}{17010}
- \frac{139\left(\sqrt{-2}\right)^7}{5443200}+... \approx 0.318132 + 1.337236i \)
For base(e) conveniently \( L=\exp(L) \)  For other bases, \( L=\exp(L)-1+k \)

For tetration base2, we first convert it to the conjugate form
\( b=\ln(\ln(2))+1 \approx 0.63348708   \)
\( f(x)\approx \exp(x)-1+0.63348708 \)
If we have a fixed point L of f(x), than the fixed point for tetration base(2) is \( \exp(\exp(L)) \)
Then the value z for the series above is \( z = \sqrt{-2\left( \ln(\ln(2))+1 \right)} \)