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 Base -1 sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 06/01/2015, 03:46 AM (This post was last modified: 06/11/2015, 02:18 AM by sheldonison.) 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) +x^ 5* (-4.3565746100039621643156264414531 - 3.7502301247100502464055812573872*I) +x^ 6* (-1.1318041242454690245256922260458 - 6.1283783467098782142226578470397*I) +x^ 7* ( 3.0061084828645889711761296470390 - 5.7823816125947263744554137445483*I) +x^ 8* ( 5.9745433797940634774673374158251 - 2.8273436649229717477537404630055*I) +x^ 9* ( 6.4072399640357260092637667875837 + 1.2767105770853541297846300721175*I) +x^10* ( 4.2784409284521602508774347288089 + 4.6443788416579634487172204079675*I) +x^11* ( 0.73158338617704415540523410346173 + 5.9398555039499918746180856218853*I) +x^12* (-2.6352841964250243277394123358536 + 4.9116891367807893917529607397721*I) +x^13* (-4.5578732532679018897237197296278 + 2.3121637552542842858973468413944*I) +x^14* (-4.5788457848699061877346163012422 - 0.61701058477634494005431989343559*I) +x^15* (-3.0659032990384675969518520419223 - 2.7570078088761801084952675435324*I) +x^16* (-0.88011420918601458586385823331260 - 3.5282720574602360239183699702888*I) +x^17* ( 1.0738785329546312953932854697650 - 2.9849683776231131178906526524810*I) +x^18* ( 2.1982752835900464315310728533988 - 1.6348712688264394747147159825565*I) +x^19* ( 2.3399559488195666090511730563920 - 0.13352187321865449953326209716318*I) +x^20* ( 1.7227439542391310359334541172193 + 0.99137050921602037784667155619193*I) +x^21* ( 0.76085978416891955471660690033717 + 1.4886323955483801392426939312910*I) +x^22* (-0.14034474661415014758633892070089 + 1.3882656297707759974928303899359*I) +x^23* (-0.71913323796305623319127293799842 + 0.90527116153812676883820919109188*I) +x^24* (-0.89959064587675680324383979225592 + 0.30961106517495655878851939684014*I) +x^25* (-0.75656748510133462107566319978234 - 0.18046113189692993221215939514617*I) +x^26* (-0.44141658773615061925620725764103 - 0.45223495248788698528420057168536*I) +x^27* (-0.10745224726450862382795580058986 - 0.49900743310206222211721872527540*I) +x^28* ( 0.13945246691591115102661207371779 - 0.38554780701128592597677894985060*I) +x^29* ( 0.25744370394997779447059695729432 - 0.20251236385971246105213308665792*I) +x^30* ( 0.25873910107198796543116044939078 - 0.028888531197355390965951961142038*I) +x^31* ( 0.18609720862424419081106403826499 + 0.088151415649246144595251737172187*I) +x^32* ( 0.088370801507306694868421922797016 + 0.13590078010761979930431907227023*I) +x^33* ( 0.0033439117348189371956724351153849 + 0.12699582548331634290066330987576*I) +x^34* (-0.049455892218686721692923486186650 + 0.085909102649729102389694142951059*I) +x^35* (-0.067548270605172667957641421984127 + 0.037015005126838298078043965623860*I) +x^36* (-0.059551576488186391894543855988881 - 0.0026474037928205663193792318045882*I) +x^37* (-0.038217040937409416477646356138421 - 0.025519241955828117368564283342434*I) +x^38* (-0.015003205220312403163117848283653 - 0.031945933627602731134761043719461*I) +x^39* ( 0.0027665041816794133045779320284133 - 0.026866018225580163159845185146197*I) +x^40* ( 0.012349597400123693848506115224522 - 0.016481929753489197026056054954405*I) +x^41* ( 0.014485298189519113498974092992732 - 0.0059295239986257272707295555021282*I) +x^42* ( 0.011723772249635138030678099442338 + 0.0017659471900015484368246587877552*I) +x^43* ( 0.0069246609726384477662995427616828 + 0.0056735542081788774751516769456914*I) +x^44* ( 0.0023016044950929365000194826308814 + 0.0063333033186449504166799862089049*I) +x^45* (-0.00093603980738322022960001461350872 + 0.0049697372450043321368279296609249*I) +x^46* (-0.0024948632946974976958630585334486 + 0.0028452876951550382895434994271022*I) +x^47* (-0.0026818555101949325018148899937741 + 0.00088346421863390101264262793133177*I) +x^48* (-0.0020533808671616156631747885834552 - 0.00044516635181834979970967270095344*I) +x^49* (-0.0011469551028080601984457349073398 - 0.0010562705145480621121314888335057*I) +x^50* (-0.00033741038804432640202585454802238 - 0.0011036749219466090304383174648716*I) +x^51* ( 0.00019611474262775267949151367155456 - 0.00082917599022233067963145950852176*I) +x^52* ( 0.00043243670165876284871239818500319 - 0.00045469485954126925413974880230706*I) +x^53* ( 0.00044262522620953510697326563013983 - 0.00012884969698913746493087958588115*I) +x^54* ( 0.00032794371636664479861859637469094 + 8.1331188326111258883644705493218 E-5*I) +x^55* ( 0.00017760389483300062145013975197124 + 0.00017175512117360801624090320420930*I) +x^56* ( 4.9360054182675212238453240092785 E-5 + 0.00017336818876939450318486622109568*I) +x^57* (-3.2034886076860068254062432265477 E-5 + 0.00012725362470064094405551878876386*I) +x^58* (-6.6346799235964233187911114104212 E-5 + 6.8443079256359286418447289196680 E-5*I) +x^59* (-6.6435597553071367039665758613702 E-5 + 1.8993032525003056441687855574967 E-5*I) +x^60* (-4.8511677830936932677576790530796 E-5 - 1.2041808787798446920958961000002 E-5*I) +x^61* (-2.6046679221224440685014079096497 E-5 - 2.4973408265950755035794323069329 E-5*I) +x^62* (-7.3364106056934194716759928959360 E-6 - 2.4942385879727308820580813901822 E-5*I) +x^63* ( 4.3281224199496364342895017742744 E-6 - 1.8188108170118220511836121335590 E-5*I) +x^64* ( 9.1727438220763464248114746243044 E-6 - 9.7942681831300162567535008174981 E-6*I) +x^65* ( 9.1846668368047518447869544422227 E-6 - 2.8389767708121607974746363979813 E-6*I) +x^66* ( 6.7119672845220409046700252453357 E-6 + 1.4864826275737812589395501761939 E-6*I) +x^67* ( 3.6401773972368767054920478392522 E-6 + 3.2909700624396166493876919208327 E-6*I) +x^68* ( 1.0974631277253078619544564693491 E-6 + 3.3201313173899023160313386975503 E-6*I) +x^69* (-4.8602204513355462724650387407186 E-7 + 2.4395208585343861481307318828241 E-6*I) +x^70* (-1.1540726634040371807450067852289 E-6 + 1.3373702865324430647506969050716 E-6*I) +x^71* (-1.1789700214940668136304702942994 E-6 + 4.2250379243771064766105693120487 E-7*I) +x^72* (-8.7368330271574051702830584236844 E-7 - 1.5001950782223013485042516661714 E-7*I) +x^73* (-4.8567632239118162782452903602315 E-7 - 3.9569123983885774310456785000304 E-7*I) +x^74* (-1.6152424966779396104183449943558 E-7 - 4.1145028296950136053654486952749 E-7*I) +x^75* ( 4.2969686709371298041093640370929 E-8 - 3.0842134596625280725673062448444 E-7*I) +x^76* ( 1.3263744582758769670734832698234 E-7 - 1.7432410540530789863466478188731 E-7*I) +x^77* ( 1.4116934398235155353625366173167 E-7 - 6.1173127012083400988841492234997 E-8*I) +x^78* ( 1.0734342321172418477390189082205 E-7 + 1.0992977287339352339741430662302 E-8*I) +x^79* ( 6.1831588893495638281639789074193 E-8 + 4.3446846621249762021124431488611 E-8*I) +x^80* ( 2.2907539574784992846643692224259 E-8 + 4.7626357235705502293728019827574 E-8*I) +x^81* (-2.2558682594750271227881662167133 E-9 + 3.6839007246004590409925792831803 E-8*I) +x^82* (-1.3893564982796096231282318965124 E-8 + 2.1668201487508227791767656645361 E-8*I) +x^83* (-1.5799704834090040959934489863961 E-8 + 8.4701096268325050434427782233371 E-9*I) +x^84* (-1.2467204614850213700664169941950 E-8 - 1.9906718640918155451027914587233 E-10*I) +x^85* (-7.5008299601753467031390322703315 E-9 - 4.3304161987170377985147910308746 E-9*I) +x^86* (-3.0894202656401301157719695071165 E-9 - 5.1533367111930139006110477456355 E-9*I) +x^87* (-1.3871202141032326502887149537226 E-10 - 4.1605983355139553596807721184966 E-9*I) +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 « Next Oldest | Next Newest »

 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 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

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