(07/02/2022, 10:37 AM)bo198214 Wrote: No, you are not missing anything. So can you calculate some more series coefficients, and show Catullus how its done?
Okay, so I can't generate the terms algebraically.
But each coefficient of \(z^k\) is the term \(c_k\) in \(c_k \log(2)^{kz}\) in the sum I wrote above.
This is done through a recursive protocol built into my program beta.gp. It's just a redundant amount of code meant to run the Schroder iteration, so it doesn't use any of the math of the beta method. I just installed this code into the program.
So this is \(A = \Psi(1)\) and the function output \(\Psi^{-1}(A*z)\) for a variable \(z\). But it gives the coefficients \(c_k\).
Code:
/*I've already initialized the base value as b=log(2)/2, and the beta polynomials, once you do that you can initialize the regular iteration, then you can run the code I'm writing*/
A = Sch_reg(1)
%39 = -0.63209866105082925035545064599078086279940391183279
Inv_Sch_reg(A*(z+O(z^50)))
%42 = 2.0000000000000000000000000000000000000000000000000 - 0.63209866105082925035545064599078086279940391183279*z - 0.22563428568113651641314342556106075219383197400950*z^2 - 0.085408173026959759449911692849916118411760709773437*z^3 - 0.033577116075467402923000166342379529173359883175780*z^4 - 0.013567533990220214760849485909118650891001851681932*z^5 - 0.0055992068394587932789325712909644497530942417856943*z^6 - 0.0023500328878460392750960673121964884395590643434851*z^7 - 0.0010000364723451153067684842123773408264572502952945*z^8 - 0.00043048070830406712985249949329081051437808143735115*z^9 - 0.00018711645867148671128315207150288305360049496894112*z^10 - 8.2011402174512465373062128600968758671644383924212 E-5*z^11 - 3.6202764736003317855243892589155487225622153451634 E-5*z^12 - 1.6080724216503292699172049221643433258042551459311 E-5*z^13 - 7.1816950016398788838199208354206545023038257355745 E-6*z^14 - 3.2227189833782963297158720669992862252425048625638 E-6*z^15 - 1.4522898416077888233911830908546717007993161862868 E-6*z^16 - 6.5692689018586711099226950941511558913933794341841 E-7*z^17 - 2.9815414068417090524975645714217900937721032122339 E-7*z^18 - 1.3573017645273591657543672816286683203861576883508 E-7*z^19 - 6.1957773072014452526396199289152844991250580056410 E-8*z^20 - 2.8352284888659532737042040264288626168300288619249 E-8*z^21 - 1.3003388847415089292648838389653585663335687232426 E-8*z^22 - 5.9760834258357606554760750999794690256238177818190 E-9*z^23 - 2.7516550155850334088613811741460857471873424553901 E-9*z^24 - 1.2691789634394696392683795976593671457297353819964 E-9*z^25 - 5.8633397392766035162414167673743212399309565423312 E-10*z^26 - 2.7127434800812277698835765166438900199344726802836 E-10*z^27 - 1.2568039997676115316158876879469486097115505664483 E-10*z^28 - 5.8301536471094220748723560631635569554785296853437 E-11*z^29 - 2.7077507782993934168201247406928073117302713586960 E-11*z^30 - 1.2589830287685641289138737053231258637289280194117 E-11*z^31 - 5.[+++]At the amount of recursion I ran, slash series precision, digit precision--we're probably accurate to at least 25-30 digits. Might have more errors as we go further out.
Either way, setting \(z= \log(2)^x\) creates the asymptotic expansion Catullus is talking about, but it's actually just a Taylor expansion/Fourier expansion.
