Base -1
#3
Here is what analytic sexp base(-1) looks like at the real axis. I've been working on a generic complex base slog/abel function program for several months, that converges nicely over a very wide range of complex bases. It works on iterating \( f(z) \mapsto \exp(z)-1+k \), where k is a complex number; and generates the Abel function for f(z), on a sickle extending from one fixed point to the other fixed point. Then the inverse of the Abel function is the superfunction, and you can generate sexp_b(z) from k=ln(ln(b))+1. I will post the code and more details sometime soon; I have a few more boundary conditions I'd like to clean up.

Once I got most of the bugs ironed out, I tried it for sexp base(-1), which corresponds to \( k = \ln(\pi)+1 + i \cdot{0.5\pi \), and it actually converged! I generated the Abel function for k, accurate to 32 decimal digits, which took ~5 seconds. Then the inverse of the Abel function was used to generate this sexp(z) Taylor series for base(-1). This solution is analytically the same as Kneser's sexp(z) solution for base(e), where we slowly modify base(e) to base(-1), from above. Both fixed points are repelling; the upper fixed point is ~=0.2660+0.2943i, but even more interestingly, the lower fixed point is -1. So at -imag(infinity), this sexp goes to -1. It also goes to -1 for positive integers!
   
Update: here is the complex plane plot, from real{-3...+8} and imag{-4...+2}
   
And the Taylor series
Code:
{sexpm1= 1
+x^ 1* ( 1.5643248936662814621073136252354 + 1.4263908193511109504199813317038*I)
+x^ 2* (-0.28772262630809147775729355647775 + 3.2371530208268087422671917275387*I)
+x^ 3* (-3.1816850373498935786354861123635 + 2.7374291225727559051245356128768*I)
+x^ 4* (-5.0636242146950889166241398735411 - 0.059878452885781276688472929090662*I)
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+x^72* (-8.7368330271574051702830584236844 E-7 - 1.5001950782223013485042516661714 E-7*I)
+x^73* (-4.8567632239118162782452903602315 E-7 - 3.9569123983885774310456785000304 E-7*I)
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+x^88* ( 1.3121979892933509568363708530764 E-9 - 2.5644007505927475914614235369619 E-9*I)
+x^89* ( 1.6520948940338556240572850856035 E-9 - 1.1108986518594313772045077308496 E-9*I)
+x^90* ( 1.3691019752450576906920860072169 E-9 - 1.1877256502881455609982203418057 E-10*I)
+x^91* ( 8.6570882592680044497740392206164 E-10 + 3.8503157128893047969315839231709 E-10*I)
+x^92* ( 3.9366240742207587380732261783442 E-10 + 5.2032003181856405684256816659319 E-10*I)
+x^93* ( 6.4174497990631189601712860239602 E-11 + 4.4416795129350336267521208755127 E-10*I)
+x^94* (-1.0870835673266687116817990083413 E-10 + 2.8852872770665232791645298648736 E-10*I)
+x^95* (-1.6086818780925010216194093476918 E-10 + 1.3745066429362638672235679117700 E-10*I)
+x^96* (-1.4203519137862846789554378177818 E-10 + 2.9388679153025891876455518626188 E-11*I)
+x^97* (-9.4920516326208743139713988999208 E-11 - 2.9218259811681215777480774083946 E-11*I)
+x^98* (-4.7284390998273022108353166361368 E-11 - 4.8771885221818007142765765828655 E-11*I)
+x^99* (-1.2291098615887457372273718360101 E-11 - 4.4755965555003292143691636361229 E-11*I)
+x^100* ( 7.3300886716904901625728612967961 E-12 - 3.0818042720312595392553918667427 E-11*I)
+x^101* ( 1.4478191613750148754299413877852 E-11 - 1.6026735055457639322378211965648 E-11*I)
+x^102* ( 1.3891096742113936146763001606371 E-11 - 4.8406113249563789457411372983718 E-12*I)
+x^103* ( 9.8727379539427057592206279444415 E-12 + 1.6454205085960430929181698364831 E-12*I)
+x^104* ( 5.3524840448256097960690813441823 E-12 + 4.1993116093661300073600159906265 E-12*I)
+x^105* ( 1.8235320448974089502909822273599 E-12 + 4.2444516790670053944185015869589 E-12*I)
+x^106* (-2.9301087149364148751828453042269 E-13 + 3.1200395791372904012030744566014 E-12*I)
+x^107* (-1.1863820896703897911141247385123 E-12 + 1.7614997859934323515218243680796 E-12*I) }
- Sheldon


Messages In This Thread
Base -1 - by marraco - 05/21/2015, 06:31 PM
RE: Base -1 - by sheldonison - 05/22/2015, 05:28 AM
RE: Base -1 - by Catullus - 06/19/2022, 11:01 PM
RE: Base -1 - by sheldonison - 06/01/2015, 03:46 AM
RE: Base -1 - by marraco - 06/02/2015, 01:08 PM
RE: Base -1 - by sheldonison - 06/02/2015, 05:27 PM
RE: Base -1 - by Daniel - 06/19/2022, 11:34 PM
RE: Base -1 - by JmsNxn - 06/20/2022, 12:40 AM
RE: Base -1 - by Daniel - 06/20/2022, 01:01 AM
RE: Base -1 - by Daniel - 06/20/2022, 03:21 AM
RE: Base -1 - by JmsNxn - 06/20/2022, 02:37 AM
RE: Base -1 - by Catullus - 07/06/2022, 09:37 AM

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