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Base -1
#1
Tetration base 1 is a constant function equal to 1, for x>-1, and has a discontinuity at -1. Or at least that's the solution if .

[Image: 3ZQkL3L.jpg?1]

The oscillatory behavior of bases 0<b<1 suggest that base 0 looks like this on the limit for b=0:

[Image: ToRZAgn.jpg?1]
We don't know if tetration has changes of state as the base converges to zero, but can only be equal to 0 or 1, so the doubt is how many discontinuities it has on a given interval.

Base -1:
, so, once , then the function must be periodically equal to -1:
This is a possible solution:

[Image: Gs1tpK1.jpg?1]
Is not the only solution; it may be displaced to the right by any real value d (0<d<1), and still will be a solution valid on the interval -2+d<x.

I wonder if there is a family of continuous solutions, and this one is the evolvent.
I have the result, but I do not yet know how to get it.
Reply
#2
(05/21/2015, 06:31 PM)marraco Wrote: Tetration base 1 is a constant function equal to 1, for x>-1, and has a discontinuity at -1. Or at least that's the solution if ... I wonder if there is a family of continuous solutions, and this one is the evolvent.

b=-1 has two primary fixed points. both repelling; I'm not sure what the other fixed points are.
-1,
0.266036599292773 + 0.294290021873387*I

Pairs of repelling fixed points can sometimes be used to build analytic complex valued solutions, with the property that tet(-1)=0. As you discovered, there also appears to be another family of solutions for 0<b<1, which is a damped oscillator type solution. So it quickly appears that there is no uniqueness what so ever... with an infinite number of interesting solutions possible.

I find it easier to work with the conjugate base, and analyze iterating the function:

instead of where

and this is a simple linear transformation from y to z

Then analyzing the function for iterating
is conjugate (or mathematically equivalent) to iterating the function
but this conjugate form is much simpler to work with and understand. k=0 is the parabolic case which corresponds to base k>0 corresponds to Kneser's real valued tetration solution, and corresponds to Marraco's bases between 0..1
And the conjugate value of k for b=-1 is

I have a series solution for the two fixed primary fixed points; http://math.eretrandre.org/tetrationforu...hp?tid=728 which turns out to have a nice Taylor series solution with and with rational coefficients. I am also in the process of debugging a very powerful generic slog/abel pari-gp program for iterating for arbitrary complex values of k. This bipolar Abel function may be unique, based on Henryk's proof, but this solution requires that the Abel function be analytic in a strip between the fixed points. For Marraco's damped oscillating solutions, the Abel function has singularities where the derivative of the sexp'(z)=0. I haven't yet generated any analytic solutions for Marraco's damped oscillating solutions for 0<b<1; that's also a longer term goal.
- Sheldon
Reply
#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 , 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 , 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
Reply
#4
(06/01/2015, 03:46 AM)sheldonison Wrote: Here is what analytic sexp base(-1) looks like at the real axis.

Wonderful!

Did you tried negative bases close to zero?

(06/01/2015, 03:46 AM)sheldonison Wrote: [Image: attachment.php?aid=1202]

This is how it looks that same curve (-1<x<2) when the real part is drawn in ordinates, and the imaginary on abscissas
[Image: B4DO0jz.jpg?1]
I have the result, but I do not yet know how to get it.
Reply
#5
(06/02/2015, 01:08 PM)marraco Wrote:
(06/01/2015, 03:46 AM)sheldonison Wrote: Here is what analytic sexp base(-1) looks like at the real axis.

Wonderful!

Did you tried negative bases close to zero?
Yes, though I'm not really getting useful results yet.

The algorithm works with
For b=0.5, that would be , where we iterate . Right now, the closest I can get to tetration base=0.5 with good convergence is b=0.5+0.39i, which is k=0.782+2.173i. I'm having several different kinds of problems with these bases, though I hope I will eventually get full convergence for some bases with imag(k)=pi, though convergence will still be limited for bases with indifferent fixed points like
- Sheldon
Reply
#6
(06/02/2015, 01:08 PM)marraco Wrote:
(06/01/2015, 03:46 AM)sheldonison Wrote: Here is what analytic sexp base(-1) looks like at the real axis.

Wonderful!

Did you tried negative bases close to zero?

I generated a negative tetration base fairly close to zero, with good precision. This is as close as I can get to Kneser analaytic tetration near base(0) so far with my new Abel function program. And its a very artistic tetration base!
Tetration base b~=-0.135335
I calculated this using my new pari-gp program (which I will publish soon), which generates the Abel function for iterating for many arbitrary complex values of k. For the tetration base of interest,

which is conjugate or equivalent to the slog/sexp base above. Call the Abel function . Then if you can take its inverse, then a reasonably straightfoward, linear equation gives you the desired sexp(z) function.



And here is the resulting sexp function, for b=-exp(-2)~=-0.135335, after taking the inverse of the Abel function. Notice the beautiful 6-cycle attracting fixed point rainbow! The two primary fixed points are both repelling, with the upper fixed point Period1~=2.327+0.182i, and the lower fixed point Period2~=2.357-1.966i The graph goes from -3 to +8 at the real axis, and +2i to -4i at the imaginary axis. I also added a similar complex plane graph to base(-1) to my earlier post; comparing the two plots is interesting.
   

Here is the Abel function, showing the logarithmic branch singularity at the two repelling fixed points,
L1 = 0.209297833082953879535 + 2.68321717005655097610i
and L2 = 1.31091514181888952589 - 1.57185702066772588857i
The Abel function is much more well behaved, especially between the two fixed points where Henryk's uniqueness proof would hold. So then, starting with Tetration base(e), this function is the same function as if you slowly modify the base, going above any singularities, and then get to b~=-0.135335.
I arbitrarily set and center the Abel function series exactly between the two fixed points. Of course, taking the inverse of the Abel function can be difficult, and it took me several weeks to get it to work. In principle, for sexp(z) you add any integer value n to z, and then find the corresponding in the well behaved region of the Abel function, nearest the line connecting the two fixed points. Then iterate or its inverse as needed.
   

And the Taylor series of the sexp function:
Code:
{sexp_bmem2= 1.00000000000000
+x^ 1* ( 1.36731941182746 + 2.03147175815011*I)
+x^ 2* (-2.16809727173656 + 4.19637865176037*I)
+x^ 3* (-7.35090243487976 + 1.04003832671689*I)
+x^ 4* (-7.76932973206147 - 7.27048735575206*I)
+x^ 5* ( 0.589449327914730 - 14.2870020608798*I)
+x^ 6* ( 14.0292409179381 - 11.7028472515246*I)
+x^ 7* ( 22.2709683173421 + 2.66099963739772*I)
+x^ 8* ( 16.2056377985895 + 21.1440445300114*I)
+x^ 9* (-4.00052427101099 + 30.5039203303803*I)
+x^10* (-27.2329501197431 + 21.4708476779723*I)
+x^11* (-38.0970145296299 - 3.63291240324545*I)
+x^12* (-27.3860956401045 - 31.1078116413744*I)
+x^13* ( 1.06541844699315 - 44.1203652438513*I)
+x^14* ( 31.9899439305810 - 33.4505698061264*I)
+x^15* ( 47.7376209547412 - 3.59516873972375*I)
+x^16* ( 38.8744926788335 + 29.6476765419447*I)
+x^17* ( 9.70089125969798 + 48.3842658339320*I)
+x^18* (-24.4212790820708 + 42.7939493910419*I)
+x^19* (-45.9099730044272 + 16.2573769961374*I)
+x^20* (-44.5098197347121 - 17.1259700424062*I)
+x^21* (-22.1847326261064 - 40.6273904231631*I)
+x^22* ( 8.85967686390925 - 43.6698842881002*I)
+x^23* ( 33.2500000163052 - 26.5678708466025*I)
+x^24* ( 40.3458148070150 + 0.768524202915067*I)
+x^25* ( 28.8345215777881 + 24.7410076380143*I)
+x^26* ( 6.16979743493568 + 34.9970724036575*I)
+x^27* (-16.1204609578251 + 28.8287241403231*I)
+x^28* (-28.3479329298633 + 11.2968326377692*I)
+x^29* (-26.7807093723794 - 8.28335004708938*I)
+x^30* (-14.3347689035538 - 21.2229505068434*I)
+x^31* ( 1.86986211043715 - 23.2002695407346*I)
+x^32* ( 14.3880364914704 - 15.3614450714276*I)
+x^33* ( 18.7338928521333 - 2.79216135431438*I)
+x^34* ( 14.7222409461250 + 8.43269295404455*I)
+x^35* ( 5.67452562183119 + 14.0254494200453*I)
+x^36* (-3.71060159812337 + 12.9117512879776*I)
+x^37* (-9.60963602162847 + 6.98400122561931*I)
+x^38* (-10.4570809778351 - 0.337640236895436*I)
+x^39* (-7.07145174988285 - 5.85226402250897*I)
+x^40* (-1.76664399349272 - 7.82595719503162*I)
+x^41* ( 2.93707465883642 - 6.33970001671262*I)
+x^42* ( 5.37101878806418 - 2.81403493262486*I)
+x^43* ( 5.16843839004206 + 0.888598888923775*I)
+x^44* ( 3.07849815118391 + 3.31057365437241*I)
+x^45* ( 0.384069596071805 + 3.86551850801680*I)
+x^46* (-1.73834230605909 + 2.83734495317994*I)
+x^47* (-2.64560166892311 + 1.03743760073807*I)
+x^48* (-2.33050443376714 - 0.651082305865240*I)
+x^49* (-1.24902489473556 - 1.63124787542959*I)
+x^50* (-0.0169840894360589 - 1.73946699221591*I)
+x^51* ( 0.868658663369885 - 1.18448619961457*I)
+x^52* ( 1.18299525936268 - 0.361586849927367*I)
+x^53* ( 0.978325940267722 + 0.350088825031553*I)
+x^54* ( 0.482889587430131 + 0.724415092792175*I)
+x^55* (-0.0365065758844401 + 0.726742005820148*I)
+x^56* (-0.384934456180744 + 0.468968975566110*I)
+x^57* (-0.489324307274890 + 0.123585934616006*I)
+x^58* (-0.387763570950288 - 0.158363645842325*I)
+x^59* (-0.180965225370035 - 0.295885561654569*I)
+x^60* ( 0.0241319422849707 - 0.285554683938124*I)
+x^61* ( 0.155188864120341 - 0.178156306182991*I)
+x^62* ( 0.189581847425479 - 0.0429500781573065*I)
+x^63* ( 0.146514704353062 + 0.0632843001040154*I)
+x^64* ( 0.0664937718450477 + 0.112603524812222*I)
+x^65* (-0.0101860812754382 + 0.106452125507408*I)
+x^66* (-0.0577722893456004 + 0.0654224809775214*I)
+x^67* (-0.0693818355508493 + 0.0155329883124371*I)
+x^68* (-0.0531079409455435 - 0.0228384982212408*I)
+x^69* (-0.0240854170285153 - 0.0402655355669488*I)
+x^70* ( 0.00325596055747948 - 0.0378718179852906*I)
+x^71* ( 0.0200340763375998 - 0.0233156379833952*I)
+x^72* ( 0.0241200757795145 - 0.00584153635301620*I)
+x^73* ( 0.0185371619345091 + 0.00751186113267998*I)
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+x^75* (-0.000740735023539575 + 0.0129058134810033*I)
+x^76* (-0.00650152175780582 + 0.00807518025914148*I)
+x^77* (-0.00799340073258151 + 0.00223883443680704*I)
+x^78* (-0.00624274443664014 - 0.00225123106804117*I)
+x^79* (-0.00301790313903449 - 0.00435347534573896*I)
+x^80* ( 4.95519208827178 E-5 - 0.00422168068371654*I)
+x^81* ( 0.00197680632972331 - 0.00271692938539455*I)
+x^82* ( 0.00253016488800373 - 0.000852950133620559*I)
+x^83* ( 0.00202974901277237 + 0.000607665530495524*I)
+x^84* ( 0.00103528430624957 + 0.00132169806540970*I)
+x^85* ( 6.59065957526732 E-5 + 0.00132686003927618*I)
+x^86* (-0.000561741464392471 + 0.000886839142800126*I)
+x^87* (-0.000765424326681099 + 0.000316850156843994*I)
+x^88* (-0.000637025595065289 - 0.000142883548386641*I)
+x^89* (-0.000345545515393005 - 0.000380277138023169*I)
+x^90* (-5.05789983559578 E-5 - 0.000400693610395819*I)
+x^91* ( 0.000148065796047292 - 0.000280347485912187*I)
+x^92* ( 0.000221173796316318 - 0.000113446239995589*I)
+x^93* ( 0.000192833811505339 + 2.64834733476581 E-5*I)
+x^94* ( 0.000111758286962626 + 0.000103362106791032*I)
+x^95* ( 2.53715103926836 E-5 + 0.000116174400636905*I)
+x^96* (-3.56048696698538 E-5 + 8.56729551567910 E-5*I)
+x^97* (-6.09270805598900 E-5 + 3.89260725733394 E-5*I)
+x^98* (-5.62341054381181 E-5 - 2.21526469386093 E-6*I)
+x^99* (-3.49179066209874 E-5 - 2.63684637269792 E-5*I)
+x^100* (-1.05945336135646 E-5 - 3.22875112016872 E-5*I)
+x^101* ( 7.52305762012355 E-6 - 2.52645938199225 E-5*I)
+x^102* ( 1.59425177720560 E-5 - 1.27707099055333 E-5*I)
+x^103* ( 1.57727846421572 E-5 - 1.10322737968916 E-6*I)
+x^104* ( 1.05156048713805 E-5 + 6.23585377877011 E-6*I)
+x^105* ( 3.94407354646783 E-6 + 8.57980000810110 E-6*I)
+x^106* (-1.25649607371861 E-6 + 7.17682194558730 E-6*I)
+x^107* (-3.93791192295531 E-6 + 4.00389923198645 E-6*I)
+x^108* (-4.24608013799630 E-6 + 8.18229284222911 E-7*I)
+x^109* (-3.04716928103069 E-6 - 1.33361911124904 E-6*I)
+x^110* (-1.34852826543113 E-6 - 2.17133005266941 E-6*I)
+x^111* ( 9.08444692415682 E-8 - 1.96010184377976 E-6*I)
+x^112* ( 9.08307425947891 E-7 - 1.19984789786628 E-6*I)
+x^113* ( 1.09392298244750 E-6 - 3.64409609844220 E-7*I)
+x^114* ( 8.48390090600844 E-7 + 2.43312052217077 E-7*I)
+x^115* ( 4.30170799802214 E-7 + 5.20103603935867 E-7*I)
+x^116* ( 4.68479223650280 E-8 + 5.13541401576740 E-7*I)
+x^117* (-1.91712000260767 E-7 + 3.43779117535491 E-7*I)
+x^118* (-2.68605100499922 E-7 + 1.34016660819384 E-7*I)
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+x^124* ( 5.79504523565308 E-8 - 9.53220412032397 E-10*I)
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+x^128* (-1.36208639341363 E-8 + 1.32938985516492 E-8*I)
+x^129* (-1.41514596475122 E-8 + 2.54467629572432 E-9*I)
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+x^131* (-4.34171589882065 E-9 - 6.93972499505146 E-9*I)
+x^132* ( 2.17584886530242 E-10 - 6.16542140126721 E-9*I)
+x^133* ( 2.73430527684382 E-9 - 3.75585929113322 E-9*I)
+x^134* ( 3.28638788849804 E-9 - 1.19158866746892 E-9*I)
+x^135* ( 2.54957434115807 E-9 + 6.38846459911396 E-10*I)
+x^136* ( 1.32019560377171 E-9 + 1.46856741870821 E-9*I)
+x^137* ( 2.06177629249362 E-10 + 1.46812045059197 E-9*I)
+x^138* (-4.86640902901193 E-10 + 1.00004882005770 E-9*I)
+x^139* (-7.21090923395241 E-10 + 4.18454531158782 E-10*I)
+x^140* (-6.23157141380506 E-10 - 4.11740694435993 E-11*I)
+x^141* (-3.70024185839065 E-10 - 2.85677342094401 E-10*I)
+x^142* (-1.10705188026581 E-10 - 3.31678654588806 E-10*I)
+x^143* ( 6.93963657594826 E-11 - 2.51980295984708 E-10*I)
+x^144* ( 1.47860601162287 E-10 - 1.27467056066238 E-10*I)
+x^145* ( 1.44645050710933 E-10 - 1.76121544604168 E-11*I)
+x^146* ( 9.69579092026992 E-11 + 4.91102655400584 E-11*I)
+x^147* ( 3.97059930309326 E-11 + 7.06141823050801 E-11*I)
+x^148* (-4.65389353106001 E-12 + 6.01633108838619 E-11*I)
+x^149* (-2.77642429168665 E-11 + 3.53084921390022 E-11*I)
+x^150* (-3.17621680179065 E-11 + 1.03698476509002 E-11*I)
}

For completeness, here is the graph at the real axis.
   
- Sheldon
Reply
#7
WOW, this is really nice.
If You post code which generates all of this, I can write C++/C/Python code based on Your example.
Just because Cis in almost all environments, while Pari-gp is rather specific tool.
Of course if You give me permission to do so.

Best Regards.
Fuji GSW690III
Nikon D3, Nikkors 14-24/2.8, 24/1.4, 35/2, 50/1.4, 85/1.4, 135/2, 80-200/2.8
Reply
#8
(06/10/2015, 07:50 PM)MorgothV8 Wrote: WOW, this is really nice.
If You post code which generates all of this, I can write C++/C/Python code based on Your example.
Just because Cis in almost all environments, while Pari-gp is rather specific tool.
Of course if You give me permission to do so.

Best Regards.

Thanks. If you're up for a challenge, once I publish what I have, we can talk about porting to a more generic language.
- Sheldon
Reply
#9
(06/10/2015, 09:20 PM)sheldonison Wrote:
(06/10/2015, 07:50 PM)MorgothV8 Wrote: WOW, this is really nice.
If You post code which generates all of this, I can write C++/C/Python code based on Your example.
Just because Cis in almost all environments, while Pari-gp is rather specific tool.
Of course if You give me permission to do so.

Best Regards.

Thanks. If you're up for a challenge, once I publish what I have, we can talk about porting to a more generic language.

Would be great, I'm not very fluent in maths, but I really like it, and I had no problems with it on High School.
My education is software developer anyway, and I can write it in my spare time.
I already tried some time ago, but PariGP have lots of built in functions, that C doesn't have.
So I would need to implement them first too.
And I DON'T want to use any external library, becaus eI want to understand how all of it works, finally.
For example find fixed point(s) of exp/log, lambert W etc. I want to write them from scratch.

Even standard pow() func from C library seems insufficient, because pow is multivalued function, and that have to be taken into account all the time.
Also it seems that logarithms from negative/any complex numbers are possible (using delta numbers etc) so standard log() is insufficient too.... I wonde rif standar + or * operators are sufficient then .... Tongue

Fuji GSW690III
Nikon D3, Nikkors 14-24/2.8, 24/1.4, 35/2, 50/1.4, 85/1.4, 135/2, 80-200/2.8
Reply


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