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A first hires look at tetration \(\lambda = 1\) and \(b = e\) - Printable Version

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A first hires look at tetration \(\lambda = 1\) and \(b = e\) - JmsNxn - 10/17/2021

Hey, everyone.

After playing with Sheldon's code, I had to slow down some of his calculations because it was creating a jump discontinuity; which was largely because he wrote,



But he chose this sum for and then iterated the functional equation. This is well and fine in theory; but we lose a tad of accuracy; so I just switched it to so we're in a better converging area. This does slow down the code a bit; but it's still leagues faster than my old code.

Secondly, I can no longer hide behind inaccuracy as the cause of the "fractal anomalies" I was seeing. At full accuracy, the fractal anomalies are precisely the branch cuts Sheldon is referring to. Now, thankfully; this is perfectly possible as a result of my paper. It was never something that couldn't happen (though my wording leads away from this). In such a sense, the function \(\text{tet}_1(s)\) is holomorphic almost everywhere on \(\mathbb{C}\). It's still to be seen if the final beta tetration \(\text{tet}_\beta(s)\) suffers the same branching problem; it seems to be a toss up at the moment. This would invalidate the final result of my paper; and I'd have to switch to \(\text{tet}_\beta\) is holomorphic almost everywhere on \(\mathbb{C}\). This would require me to pivot towards the asymptotic thesis; which is these are asymptotic solutions primarily at \(\Re(s) = \infty\).

Sheldon, believes these functions are nowhere analytic on \(\mathbb{R}\); I still am not fully convinced of this. I agree this case does have branch points on the real line, and in the neighborhood; I'm not convinced they are dense yet--especially by the below graph. That being said, if they are analytic; they are a very ugly analytic; with taylor series with horrible radii of convergence caused by points of non-analycity on \(\mathbb{R}\) and in the neighborhood. I especially disagree with nowhere analycity on the real line because at \(\Re(s) = \infty\) we can define an implicit function which is analytic--and showing \(\tau\) is this implicit function isn't too hard. So it may not be the best kind of analytic for \(0 < \Re(s) < 1\); but for \(\Re(s) > R\) I see no reason it won't be. Again though, with horrible radii of convergence.

I've tried to add Ember's protocol of catching underflows/overflows; but I don't think it's possible how he's describing. So instead, any overflow is set to 0 by default. So if it's too small, it's zero. And if it's too big, it's zero. This still helps you discern the shape, but not exactly.  This is tetration \(\text{tet}_1(s)\) for \(-2 \le \Re(s) \le 8\) and \( -5 \le \Im(s) \le 5\).

[attachment=1640]


You can see the branch cuts appearing in shifts to the left of where Sheldon's zeroes were found. So if the final beta tetration suffers these same zeroes; we can expect the same phenomena. At the moment, I have no obvious way of adapting sheldon's code to the non-periodic case; but I'm working on it. In that case we do not have the benefit of a exponential series; so we have to be more clever.

 The more I've been playing with this though, is that if \(\text{tet}_\beta\) is as nice as I said, it should be Kneser. This is because as we decrease the period the function \(\lim_{\lambda \to 0}\text{tet}_\lambda\) satisfies, on \(S = \{s \in \mathbb{C}\,|\,\pi/2 < |\arg(s)| < \pi\}\), \(\lim_{|s| \to \infty} \lim_{\lambda \to 0} \text{tet}_\lambda(s) = L\), our familiar fixed point. And this is equivalent to being Kneser per Paulsen & Cowgill.

You can also new clusters of singularities too. I believe these happen on a bunch of petals, not just the ones Sheldon chose. Unfortunately I couldn't see how to adapt his code to locate these other petals accurately. So the first batch of petals are most noticeable, but you can notice a similar phenomena happening elsewhere.


Now, in worst case scenario; this is exactly as Sheldon said; and then, we're at a point of question: "Can we salvage this?" For this I've been mulling over Kouznetsov's approach. Say we take \(\tau^{100}(s)\) and we limit the equation:

$$
G(s) = \lim_{n\to\infty} \exp^{\circ n}(\beta(s-n) + \tau^{100}(s-n))
$$

We may be able to still derive an analytic function. Particularly; it'll still be periodic--and by definition will not equal \(\text{tet}_1(s)\). This should converge (? not certain)-- and if you graph \(\text{tet}_1(s)\) way off in the left half plane, the branch cuts quiet themselves to nothing. It may not work for \(n=100\) but a clever use of \(\tau^m\) where \(m = g(n)\) may work. So all is not lost yet!


Anyway, just thought I'd give a quick update on what I've been working on. I'm just gonna fiddle with the code a bit more before I upload it. But it's looking much better.

Regards, James


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - Ember Edison - 10/18/2021

Sorry but Where is your code Huh


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - JmsNxn - 10/20/2021

Hey, Ember!

I was just double checking everything before I uploaded it. It's about as good as I could get it. Here it is!

[attachment=1649]

To begin you just run init(l,b) where l is the multiplier and b is the base; where I've opted for the exponential to be \(\exp(bz)\) as opposed to \(b^z\). So it's the log of our usual idea of a base. After this, mult = l and base = b and all the code will run with these  values.

I've set series precision to 300 by default--this will also tell init to initialize beta with 300 iterations; you can tell init to do more iterations by writing init(l,b,count) where count is as large as you want.

There's the beta(z) which is the beta function as we know it, and there's beta_N(z) = beta(z-1/mult) which sort of guesses about where beta(0) = 1; which can be helpful.

Then I have tau which is basically as before, I have tau_N which does the same thing but with beta_N (effectively we get beta(z-1/mult) + tau(z-1/mult) = beta_N(z) + tau_N(z)).

Then I have Sexp_N(z) = beta(z) + tau(z)

Then I have some normalization protocols which I have not made perfectly. So use these are your own risk. I suggest avoiding them because I couldn't get them to work efficiently.

Then I have an error catching Sexp_Chop; which runs Sexp_N but assigns overflows to 0.

And lastly we have Mike3's graphing program.


It's still not working perfectly; but you can make some interesting graphs~!

Remember, if something isn't accurate enough; especially for exp(b) in the Shell-Thron region, you should increase the iteration depth to 1000 rather than the 300/100 I have it defaulted at. All functions which require a recursive process can always be increased adding an argument at the end telling us to go deeper. It'll slow down, but still works.



For example for base = 1/2 log(2) and mult = 1 (which is \(\sqrt{2}\) with period \(2 \pi i\)); we should increase the iteration size, displayed by this series of commands.

Code:
\r beta.gp

init(1,1/2*log(2))                  
%18 = 1.564261222210131790778549754460129428491876323129948191002835819035613273620634642498895851710066100903308133764738206355851478602526882185265925756641387118119094998243398830038599470896730999483543814425736970911144963070281914026857086826620357221541090113816887067281078711614531339222156998081447309790244319545423747866146567264533076447322117620483915175231174366379487441616911454122848179199112122759500174168803573509295856827663681833419381828025553381256620789325553560541892825360032209299559658330609591389027881708000951260708812941422980385032861100916859573318961083581807428528182941976294925218221931864653748148004212542254033819429530318333995961761644719979331349429828636607511364571319261147596784340347107762265797198610630616500412557659160530271569572410889273239386768755588383041477288488811874731173091849489782976321403004177787752092145657932236775285773138327986291591592609383932128157791680437486548324607143686079099728339143938961640299350007536653878755905191383 E-131*x^301 - 4.25210285529693790527285065730685566423856927884338819904671846018842502361777696312451939992828810846562035284285926394784958660130683312053366473488947159390779848379882423303752137192022859516295323642765662525146035309101258638630770413909736146293981604200724794713896084891832054480051256318656741860459128181083783798986278824158861233454902515606093835122406648120763847088632685421088193923678498772915198635279728440778432699242076793700148037492404851592545236134114062366606067763292167420239564631097763785673942160258571537034559977589295508953464012090592239323442023391849716699984449086891994099430629712973104765071472398242891910196951719119063709188905719229300048443937538808519146854385962753001297421400734961019460525662304690087212722407012979278965146856275695478314776002731652345905568813916198114309722509268786956901596197449718222114147676306116143692714485832928245746422734489712068280247352741023716908721658129638981348817814145897669767[+++]
/*initialize a taylor series with series precision 300 and digit precision 1000--and iteration depth the level of series precision; 300*/

exp(base*Sexp_N(20)) - Sexp_N(21)
%23 = 6.737597502028215895398620843696236117166364306438656730914692116631654872544255366451462578322003495976805126837890712397919459344514774919917508681165472836749210533903165948317745570373902391482684501540809383208322394059430181521248811175106872811147313464748579923692322028975089878286946631661798352729250155962380480386588801345611903110016812773184708048467195829968925227071197321226856514192835405255737908428990186663447493984701568953727021229903841439184400762305040795591897890343823077710282717643654799766824289427284892197695223945749475307358085395568298362956491434997806409729364223915756790083078300989399923024250388273834525408237713835620759412604059099970462993780019648963671928534089764873244635876267951224452935124979344793492129637473690923720779127195890174728413782042970011778575650217541396581866318160653584491379862020798094032665500075919122443813503166413270802019551162748562763296218553632123685926882341890037349937874415040577000264990059647 E-37
/*only 37 digit accuracy*/


exp(base*Sexp_N(20,1000)) - Sexp_N(21,1000)
%24 = 1.66171450001862563106404933114559505976485524171097378938090272158872007655879489367361586291957952497122358400290479795017486957496561408052819418173382190947343996592382708256286967388352330654947498944666671624211474620041854493853593613507116453663434430882649249952404580301675735241411577059169080541753011640861540888215334710980484257796870386621407891157469581822495578011230457960009887416702323209239718682420278604984976190997046976889281026599531814190413891776554964778698536479272100621498248789245725061768018728557071393481202535221587679833897909792732553415221205379580734473388977131011282079206469925798527095296896517035896563512705029689274313122285362234704248058642947281054062453790199008676064131931708352 E-284
/*284 digit accuracy by increasing the iteration depth*/
I'm currently making a big graph to see what this beast looks like....


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - JmsNxn - 10/23/2021

Hey,

I made some big graphs (more are on the way, they just compile really slow). I've stuck with the convention that \(e^{bz}\) is the exponential and not \(b^z\)--this makes it a bit easier to evaluate.

Here is, \(\text{tet}_{1,\sqrt{2}}(z)\) over a large domain. This is tetration base \(b =  \log(2)/2\) with period \(2 \pi i\). I kept it to 100 iterations, which gives about 37 digit accuracy.  Again, I've coded overflows to zero.

[attachment=1661]

Everything is very regular and smooth; interestingly enough, the essential singularities are very quiet; they don't really contribute much. I think it's because we are using a bounded base.

I thought I'd take a crack at a complex base too. So this is tetration with base \(b = \log(2)/2 + 0.3i\)--which still resides in the Shell-Thron region; and \(2\pi i\) period again. I'm betting we get something just as nice for all values in the Shell-Thron region. I kept it to 100 iterations, which gives about 28 digit accuracy. Again, I've coded overflows to zero:

[attachment=1662]

I'm currently drawing up \(b = \log(1/2)\) and \(b= 1/e\) at the moment (recalling I treat the base as \(e^{bz}\) and not \(b^z\)), but I couldn't make all these graphs synchronously because more than two graphs at once slows my computer too much.



This seems to imply that the infinite composition/beta method is leagues more effective in the Shell-Thron region.


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - Ember Edison - 10/24/2021

Trial report of the last few days

1. The new beta_taylor is indeed very fast, but the problem is that at base=1E5, I*1E5, the new beta_taylor seems to be completely wrong.

Please check base=1E5, I*1E5, -1E5, and preferably accelerated to 1E13, 1E-13

(Now in the process of drawing 1E-5, it also seems abnormal)

2021/10/26:   I retract my comment so as not to mislead the public. Fortunately, I wasn't too far off the mark.

2. beta(z), is very fast. but Sexp_N(z,20) is slow, so normalization can be as slow as about two hours, like old beta_taylor

3. normalization in Shell-Thron region 100% fail
I will not try this feature again and will focus on drawing
Although, intuitively the smaller the base the more it needs to be normalization, but the success rate is too low

Some of my changes[attachment=1667]


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - sheldonison - 10/24/2021

(10/23/2021, 11:47 PM)JmsNxn Wrote: ....
Here is, \(\text{tet}_{1,\sqrt{2}}(z)\) over a large domain. This is tetration base \(b =  \log(2)/2\) with period \(2 \pi i\). I kept it to 100 iterations, which gives about 37 digit accuracy.  Again, I've coded overflows to zero.
...the essential singularities are very quiet; they don't really contribute much. I think it's because we are using a bounded base.

Hey James,
A \(2\pi i\) periodic solution for \(b=\sqrt{2}\) that is analytic and converges would seem to be quite the "cool" thing.  Do you think all of the essential singularities are at \(\pm \pi i\)?  Are there any spaces "between" these singularities, or do they get arbitrarily close together?

Assuming this solution is analytic, then there should also be an analytic 1-cyclic Fourier series connecting this solution to the standard solution from the fixed point of 2.  This deserves some investigation, probably not till next week though.

What if you used a Beta solution whose periodic period is greater than the \(\frac{-2\pi i}{\ln(\ln(2))}\approx17.14i\) period?  Would the fixed point of four show up at \(\approx\Im(8.57i)\), like it does in the conventional solution?


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - Ember Edison - 10/24/2021

2021/10/26:   I retract my comment so as not to mislead the public. Fortunately, I wasn't too far off the mark.


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - JmsNxn - 10/25/2021

Hey!

I'll address Ember's concerns later tonight (my best guess is that you should initialize with a larger iteration depth, and add more terms to the series); and I'll do some tests with a larger period of \(\sqrt{2}\) for Sheldon.

I just wanted to quickly post some more graphs. This is for \(b = -\log(2)\)

[attachment=1671]

And this is for \(b = 1/e\)

[attachment=1672]




Quick update:


Code:
\r beta.gp

init(0.3,log(2)/2,500)            
%18 = 2.631096458622487533105232674332282321448030524841955893353502131514783001693269821395505376936048771562394003877698091183948664395503950719728846995192648240203544746340314048646010273735046291935196447187205197891833410441289897077722816061114574614833820726863519318709419123543937011250057653793313045829282898107397477075103258574939941723388723994967234082096494933055445889786376836670109965964856320731189118240930373227297559517898874450268827432478688622532519326328434535519902544400370535282435395821380154235083221984302198447692568782203961738103919630922937791824800330282245562973498018308523024032096425898239456533257333665783308056501418795810150614147985596084632112448875082019921952766370395725474256550516524405347456420908547124344401325959174335034457762909786158078516732668354654492118136268481379068299711375401508050365205973452994190052522777830795171627139536312574499186117911301251845620498210685451865585871630207402792466855481092330640963332394836791806714011826501 E-66*x^501 - 3.551608728253595611856992486866354231935644548617339428782544749165111641670135678286607591528745865709888943728866606599043785912191090317610385202615324419336808917224921715155621942329560763328537591602432786193428489669819776062644322181081577454680941375493838623380803472953962742842726699575513599825199063403079349021529656789967456653267328092950728075807360119559456377382360484239150491147929831731312248046142278554095557336706118502745368493508061026962242646395504763702494204895868428711474537318225158830603954868577666333308744139067712704148309370221539879773922630051684300138939163387395463361442263559726221004462619641965340490647278284714387726602256224293848926996915219592081672291108929419888740346443257848376798192882821094646609446930738765178576510745669827317290891863621108521232847370796686966725306324613525050941197156930964370126438136717756524147118195370600983652943572721469113764074807254719136666012343113103979899230082947560255071[+++]

beta(1) - exp(base*beta(0))/2
%19 = 0.E-1001

beta(3+I) - exp(base*beta(2+I))/(1+exp(-0.3*(2+I)))
%20 = 0.E-1001 + 0.E-1002*I

Sexp_N(8.57*I)
%21 = 3.999706578082077749667519088239874818457068382888025288693853120133488946393614418201645725102442014092576923221024443376471419942364668796395803908785526738240778062929719298841796957526587565049732889544485044534754733642315154942850182981625655457253805112552151892337520549269311557444341155015995863683684418262299183469233895833797714435999575746795028563095929476238122625874557355853526627606020414478257555557343548924115505421226082999959122482034968924799489255076503343643170779195951524522266609561477137406927033934100918933261558751726142422181650512047937874821400709667986890709225702893596512607881102983605616184694257652120929543100636557296195613735984226923083755862952597448535653336137777306556028092131688101363655212818717279089669730864191713814026860103590563768033399582848655870356519532662995230548571718980135191493838074013816426328947417521073317168584580071771535510106114715944141560246454318770428936985958956520554979520538165555864753664993377530126784943431371 + 0.00016399098349320826274344285773639576466432323053935938101282266586748104908489413418450033227869442673308967030910103025913297888850066784202994025079855245463790879230751569897404878723740662367631569911221429137554122186155482395023245562769064405072787483925649179232448064220188171154785763712415223342269534226015647014767276085435940775657164807001626711574503604030163557890775952363406618798393637734778549372380802838043961537819418247101883807970339880261032065838324326996790136059660463283001587548264474032123874713064683241508027661113872075732230499109682429232691890537542213111382036974048668744537688301371087124445852304764592519460889959088545279634686582440912180582548822946712276932141410533071373347085658813754967128005135289329032678412527975578994706152800185482083338182203427003286526440938309963162050281733939869501283429775674100188019955928635849513033693464077500816089572180569769608950078087973609269514599260766463417004549366061052822813064445[+++]



So yes, Sheldon, the fixed point 4 seems to be appearing for period \(2\pi i / 0.3\) at about \(8.57i\). Fascinating! The entire solution seems to be hovering about 4 too; I'll have to look into this. The best guess I have is that once \(\beta(z) > 2\) (by growing the period) taking successive \(\log\)'s just shoot us to the fixed point 4 very fast.


Ahh! I can get a better estimate. The value Sexp_N(z+36.5) is normalized to two digits and at here, if I write Sexp_N(8.57*I+36.5) we do not aggregate towards 4.  It appears way off in the left half plane we are in a neighborhood of 4; and as we push forward we get closer to 2. I'm making a graph as we speak; to see if any extraneous singularities appear.

It does not appear to produce anymore singularities than the ones at \(\pi i\). We seem to have a zipper effect, though. Where there's a whole lot of overflows very very close to the essential singularity (so its very possible we get a similar kind of singularity wall near these)--but they are no where near as prominent as with \(b=1\).  If I had to guess, there are probably branch cuts for \(|z - \pi i| < \delta\). The rule seems to be, as it explodes it gets too close to zero and creates a cluster of zeroes. For \(b=1\) it explodes almost everywhere; for \(b = \log(2)/2\) it seems much more rare; and only explodes near the singularities; and only really really close to the singularities.


As for your concerns Ember, I'm working on seeing if I use 1000 iterations that this solves the case for \(b \to -\infty\) and \(b \to \infty\) (I presume you are saying about 1 everything seems to be fine? I'm testing this either way. I think the key is just to run more iterations; as we should require far more iterations for these cases either way because \(\exp(b)\) is too small and doesn't contribute enough in each iteration, or \(exp(b)\) is too large and we flatline without gaining enough data; and \(b \to 0\) we definitely need more iterations because \(\exp(b)\) needs to multiply together a lot of times to be significant... I'll look closer though and see if I can get it to work.

And just to double check that you're calling everything correctly; recall that \(exp(bz)\) is my use of the value b in the program, not \(b^z\). I know that's confusing; but it does make for easier calculations; maybe I should call this the  log_base as opposed to the base...




Alright, so I see it working fine as I've written it if you do 1000 iterations, ember.

If you write:


Code:
init(1,log(1E13),1000)
%19 = 0.1298145027662377249554599157008877151682143003128644524757079278979362913866331797271701888068439189029209872646053124282157380258921547849503322680507451036011734448253090944559927527910545286015333130026970194332099683907825169797692000972496119598542982846706547311033815441682636620435312863262339908641392125140239246855244370238577057554318278553533344885496257772890926045914841936765733100350171996705751578801989804726133956504302479084323032007236624920688356199425463963007349873891702994687923301634644245859981765881732250210450892897055360680990897102889622121645463886586668151782613732020124539022262802611492957416960566976482150034972447346698347063745676475718520365761009248398644046500678573881216772113382666947611019405347879455445700750287079415002136218255589892403345359942280998188409417125477168811268463839261875676275872368827518089145095330990834954664787696406347928023695805125887443616425438846814665958822805420863277174567671590079176777470528316066082659636306803*x^1001 + 0.147473653053442865631157035427473875080867865829832534094307842262273160987865472789667798585872826705028909719280552151852480469856885972279146254360570603777109162529674839399414745828816933297154150029365357444694185877927574508812875030346865369823581099951894993399184536345315903822915508614797745973254831399717431826306771663307990820071325906154126433059687692945379394787638033500104486234170774527617199550685623554547057368964512626396057683789790726475291903574750343331728182074147135700498493931918915756219722892991984603849846691930385990580664095031201519801733037446841275199193311597043081561480021200639844254242094171930091932297620720336275574979761869970372424005743363470983026536013253019058536121928933420055121589855817764539236186515019925812179793469487936601288843002536394612391253572140129223915057964939964708246737777381152266543649543321701843605237792859647703079607066550372486881891851261130440387687071195091632534917545010042549125092[+++]

beta(0)
%20 = 1414450995945612060947115.320174756681858078057874605652777604893643857656899842844600963717877938156097005878927467624521771990046979564389001621458201307316280575942560885360833844103595073073467887931720315063957621577960489707592328361073821042747896857384772894295035129405458386859341578335548278492081505414799604926955500433383087653381943317601573349313619420005980289805175795845442890404224597322634803403014064158487550885342664377516178216641103991599046006266954512534337714831890884177315822337844298606004186426436051863650453955794783406851318828012079460937975363304619581521101908374132971855046068482249733595123149701247643351796165168357974007223011462677977535292905116259916897495207299586836827826381156556015893685614975389428591552177079245736865258045146262847552724116202126832916282670023587016096471516361895327419549062108248641702076730291867947132230197926184667100688309716636020900211538440589676900988142832395844445552122643741398290903668030316337825122762880688

beta(-1)
%21 = 1.901610016152517836127978333469384205313848906110750808508094436363114688118758605358383118047198930845250681200594120723639141038149107248709281697818058094157264168742309700134388265985357474781592768149778341026299059439784731120149254674959434807295352399322117521569249845868303777151841979920986302517212340090907787534681559432021598636376148785197130397225739780254441199518519742289277107591288431848424980010732274932098939724753124093484522901900359158447771190590225203656901236806994876625732865993018071615444654877883756429033856822812849918891732498892785952037697288153912660327603225963166736773060193081162071736491999611146989321643134859072697118463657038049726913351676697579063292119650620280956874815282245805922922401390587471049844997297665467747849653792639715692709789309451359813102461713401071296247253658303924317304914585609581499805528267876819983247380405155254744063134279789791048897579173416734162054688213666230778813995514168016909704910423171274534392751519390

beta(z) - exp(base*beta(z-1))/(1+exp(-mult*(z-1)))
%23 = 0.E-977 + 0.E-975*z + 0.E-972*z^2 + 0.E-970*z^3 + 0.E-968*z^4 + 0.E-966*z^5 + 0.E-965*z^6 + 0.E-963*z^7 + 0.E-961*z^8 + 0.E-959*z^9 + 0.E-958*z^10 + 0.E-956*z^11 + 0.E-955*z^12 + 0.E-953*z^13 + 0.E-952*z^14 + 0.E-950*z^15 + 0.E-949*z^16 + 0.E-948*z^17 + 0.E-946*z^18 + 0.E-945*z^19 + 0.E-944*z^20 + 0.E-942*z^21 + 0.E-941*z^22 + 0.E-940*z^23 + 0.E-938*z^24 + 0.E-937*z^25 + 0.E-936*z^26 + 0.E-934*z^27 + 0.E-933*z^28 + 0.E-932*z^29 + 0.E-931*z^30 + 0.E-929*z^31 + 0.E-928*z^32 + 0.E-927*z^33 + 0.E-926*z^34 + 0.E-925*z^35 + 0.E-923*z^36 + 0.E-922*z^37 + 0.E-921*z^38 + 0.E-920*z^39 + 0.E-919*z^40 + 0.E-918*z^41 + 0.E-916*z^42 + 0.E-915*z^43 + 0.E-914*z^44 + 0.E-913*z^45 + 0.E-912*z^46 + 0.E-911*z^47 + 0.E-910*z^48 + 0.E-909*z^49 + 0.E-907*z^50 + 0.E-906*z^51 + 0.E-905*z^52 + 0.E-904*z^53 + 0.E-903*z^54 + 0.E-902*z^55 + 0.E-901*z^56 + 0.E-900*z^57 + 0.E-899*z^58 + 0.E-898*z^59 + 0.E-897*z^60 + 0.E-896*z^61 + 0.E-894*z^62 + 0.E-894*z^63 + 0.E-892*z^64 + 0.E-891*z^65 + 0.E-891*z^66 + 0.E-889*z^67 + 0.E-888*z^68 + 0.E-887*z^69 + 0.E-886*z^70 + 0.E-885*z^71 + 0.E-884*z^72 + 0.E-883*z^73 + 0.E-882*z^74 + 0.E-881*z^75 + 0.E-880*z^76 + 0.E-879*z^77 + 0.E-878*z^78 + 0.E-877*z^79 + 0.E-876*z^80 + 0.E-875*z^81 + 0.E-874*z^82 + 0.E-873*z^83 + 0.E-872*z^84 + 0.E-871*z^85 + 0.E-870*z^86 + 0.E-869*z^87 + 0.E-869*z^88 + 0.E-867*z^89 + 0.E-866*z^90 + 0.E-866*z^91 + 0.E-865*z^92 + 0.E-864*z^93 + 0.E-863*z^94 + 0.E-862*z^95 + 0.E-861*z^96 + 0.E-860*z^97 + 0.E-859*z^98 + 0.E-858*z^99 + 0.E-857*z^100 + 0.E-856*z^101 + 0.E-855*z^102 + 0.E-854*z^103 + 0.E-853*z^104 + 0.E-852*z^105 + 0.E-851*z^106 + 0.E-851*z^107 + 0.E-850*z^108 + 0.E-849*z^109 + 0.E-848*z^110 + 0.E-847*z^111 + 0.E-846*z^112 + 0.E-845*z^113 + 0.E-844*z^114 + 0.E-843*z^115 + 0.E-842*z^116 + 0.E-841*z^117 + 0.E-840*z^118 + 0.E-839*z^119 + 0.E-838*z^120 + 0.E-838*z^121 + 0.E-837*z^122 + 0.E-836*z^123 + 0.E-835*z^124 + 0.E-834*z^125 + 0.E-833*z^126 + 0.E-832*z^127 + 0.E-831*z^128 + 0.E-830*z^129 + 0.E-830*z^130 + 0.E-829*z[+++]


The polynomial is accurate to about 1000 digits minus a 100 or so digits as you go further out in the Taylor series. So this works fine. Note, that for \(1E13^z\) we have to initialize \( b= \log(1E13)\).

Checking \(\exp(1E-5)\) everything seems to be fine too,

Code:
\r beta.gp
init(1,1E-5,1000)

%18 = 1.867330055199129483817007152110538060781215935289567548234021463168942214723070952518319438765136613995403148935429902384258796625604333525563289395831897344706412053257973078121074881788660653110056171469736600486189406493985861728687934327453061647205840843094381128273217297109896601886658152751270767897513317591536932820453332761119779087095385157289868666305206302376996104361749519689201347997796384813943245756218945129133013164573495328161655118001777903653222494402360106007627775757019651884142637666626524017922341017710344216523730233489789676656227987571164132924764426276951805170446725900830952225016751560028566276355701321607068298293997321006004129750152072144226069346654483291732435339995068922311145155201918512862496656560034151360645770804625885865552651243708886029803146201257801121545404880854114275019620107059968046945842422469617448684073297438526723880689324239609758630715608589289084176161224591052772118525851736242935331821866005175315608471224288553113244292534633 E-435*x^1001 - 5.0759293567832195619630136083908362475908694776195837301533141912936637457889786758986257116260422265063452260171480107148204940963719051340648424441435035678474814977882146675734690588594635177004047014910270176928768838597542076461604833887052188874336490947650419530024592679966528366142117704308090674550191865200147309588323391551062728866285304303543590278770416374807577796752986651982662441295486086053446184784321433604141123144761951206842909395418418039391966341626903834265909429323286332225048527564831163677048518326981444823215096164651843795503620764033806177240603813334343666068633793936666571711501844239447970306093006548282895751750809795318232072502434289358633230094085741399410799370763562817122164707528223796034289907910768546277410484589018556115620693341512349928888062772793109530446328256037828689134132576576692095763686217833484052211485062751436848109509509644671469840883707760349093315928095502131386138569227559129375366790113028450630[+++]

beta(z+1) - exp(base*beta(z))/(1+exp(-mult*z))
%19 = 0.E-1021 + 0.E-1021*z + 0.E-1007*z^2 + 0.E-1022*z^3 + 0.E-1008*z^4 + 0.E-1023*z^5 + 0.E-1009*z^6 + 0.E-1024*z^7 + 0.E-1011*z^8 + 0.E-1025*z^9 + 0.E-1012*z^10 + 0.E-1026*z^11 + 0.E-1012*z^12 + 0.E-1027*z^13 + 0.E-1013*z^14 + 0.E-1028*z^15 + 0.E-1015*z^16 + 0.E-1029*z^17 + 0.E-1016*z^18 + 0.E-1030*z^19 + 0.E-1017*z^20 + 0.E-1031*z^21 + 0.E-1018*z^22 + 0.E-1032*z^23 + 0.E-1018*z^24 + 0.E-1033*z^25 + 0.E-1020*z^26 + 0.E-1034*z^27 + 0.E-1021*z^28 + 0.E-1035*z^29 + 0.E-1023*z^30 + 0.E-1036*z^31 + 0.E-1023*z^32 + 0.E-1037*z^33 + 0.E-1024*z^34 + 0.E-1038*z^35 + 0.E-1025*z^36 + 0.E-1039*z^37 + 0.E-1026*z^38 + 0.E-1040*z^39 + 0.E-1028*z^40 + 0.E-1041*z^41 + 0.E-1028*z^42 + 0.E-1042*z^43 + 0.E-1029*z^44 + 0.E-1043*z^45 + 0.E-1030*z^46 + 0.E-1044*z^47 + 0.E-1031*z^48 + 0.E-1045*z^49 + 0.E-1034*z^50 + 0.E-1046*z^51 + 0.E-1033*z^52 + 0.E-1047*z^53 + 0.E-1034*z^54 + 0.E-1048*z^55 + 0.E-1035*z^56 + 0.E-1049*z^57 + 0.E-1037*z^58 + 0.E-1050*z^59 + 0.E-1039*z^60 + 0.E-1051*z^61 + 0.E-1039*z^62 + 0.E-1052*z^63 + 0.E-1039*z^64 + 0.E-1053*z^65 + 0.E-1040*z^66 + 0.E-1054*z^67 + 0.E-1042*z^68 + 0.E-1055*z^69 + 0.E-1043*z^70 + 0.E-1056*z^71 + 0.E-1044*z^72 + 0.E-1057*z^73 + 0.E-1045*z^74 + 0.E-1058*z^75 + 0.E-1046*z^76 + 0.E-1059*z^77 + 0.E-1047*z^78 + 0.E-1060*z^79 + 0.E-1048*z^80 + 0.E-1061*z^81 + 0.E-1049*z^82 + 0.E-1062*z^83 + 0.E-1050*z^84 + 0.E-1063*z^85 + 0.E-1051*z^86 + 0.E-1064*z^87 + 0.E-1052*z^88 + 0.E-1065*z^89 + 0.E-1053*z^90 + 0.E-1066*z^91 + 0.E-1054*z^92 + 0.E-1067*z^93 + 0.E-1055*z^94 + 0.E-1068*z^95 + 0.E-1056*z^96 + 0.E-1069*z^97 + 0.E-1057*z^98 + 0.E-1070*z^99 + 0.E-1059*z^100 + 0.E-1071*z^101 + 0.E-1059*z^102 + 0.E-1072*z^103 + 0.E-1060*z^104 + 0.E-1073*z^105 + 0.E-1061*z^106 + 0.E-1074*z^107 + 0.E-1062*z^108 + 0.E-1075*z^109 + 0.E-1064*z^110 + 0.E-1076*z^111 + 0.E-1065*z^112 + 0.E-1077*z^113 + 0.E-1065*z^114 + 0.E-1078*z^115 + 0.E-1066*z^116 + 0.E-1079*z^117 + 0.E-1067*z^118 + 0.E-1080*z^119 + 0.E-1069*z^120 + 0.E-1081*z^121 + 0.E-1070*z^122 + 0.E-1082*z^123[+++]

beta(0)
%20 = 0.2689417419563710366039719472062873185215020928828650674321273582658311568854302231048803149537725115669719644592365691206024833703045301897804905756961127940104712047578049180583976391795876503241764058281940782410310734537112064321154976730554798700952031985773854188740355169461834265977900484617254533553336430584117575930526696609993354791202354778229856674615022593482430368177769171363496160987180732750808086063903949784337778283041175541285255317781996894878796384571813251925774234953035560127138989095781243264371493256906396384217070715231641943620520770500995971026312972865815795790139999797010542197134103981965377264006463739392027985985185642276544688206021511426727865321224504696377093331210970273315994516193548262696720420057000427947393581818941111820791134295482029254004345757705541869723534080784223288326684428309580168593254754227394719873458643186109020492843037719079618914694627511607000439817514803588350166554987436033554204857212529991108599169640595492976506859763207

And also for \(\exp(1E-5)\) this seems to be working fine, where I've done 1000 iterations again.

Looking at \(\exp(-1E5)\) everything seems fine too, 1000 iterations seems to be the key.

Code:
\r beta.gp
init(1,-1E5,1000)
       
%18 = 9.466881219443518654605370147335949951251837259112187304427745580057432085900873327044139987840663420717783316386290363754656360091342015745092657075911673953262784855213003432585717767942192848902012409484641595920316001151496059574348580665643124765947898493918040650796054455164878618494141749427861236237647398475914155233265601769580250450752576189625338512346513114902606577393235755215036652953055732628116966118989902211831105973626759837602361829038061000228316257010769274960918786321323103138035798512099608429710738181456487098350868520520356438400959339348666900964072585498257771579322568240201530189609307769620004289329749721385782802956381701750213197734241524077054275514604162865490052381917671158035390458000415800630560344705521593734128583470360203765691622573521637742380997943971167584571327292934521761349537602186646384781324780674339800153469113186324721678548032659213522717766812469724848364899424773262678589242417640441864999075477682356078027647351416626681357007313160 E3544*x^1001 - 3.0599953829150688142081242293249888303681584520093663493304439032593179049702875520285452858127147769623184230606469865656375242352479521819403986461698183551076746501539188781221340157735139692402945464989369117607242279733242347391047439188365626082255580048878029177404303035563562380100627597857145478096762915287687601054126071440356506237922183349324127889517739983338966225707203360854361847206752806538514031046514069107827715556229651500460132657334538161709421262347279975925950349908506010871615652871738112399403748268151394423455468182929984351904268668098998041319497812725269922650660393909620981552212901211014036588208344003260113617140154571700876807787903101084575233281192606904915472959432894461210242502091654708909054825573158865441834166136945890528378377472525297584952932241862057413663015312321487787178325466119722395772576688807421855484290847685951771036960626255281005860136883947858659135520087164847526896019219977701047392229656829285362[+++]

beta(0)
%19 = 0.2689414213699951207488407581781637256348553598349434807236340920809595469297953606125254679240187547078255335068592254335588672690652010520214276941920607183354336479166115304013137160346148649813492756957003687952091081940535504266480986153893321591085761333389050293676059934603293561309628998999520352121612007828243337426386436001987833121463197736652484862552352223417798474659899584614026622157472965450918158838274968608551169827677953740491169754122846568846384258169856132825812401694860612081747698667812435006064342201362165251624951611544121351796968796669662868138541682614605142088833395610168874213350871482822493046910113973156096096554920649461710601929290740963614676083778709036887742869953146280360183027838514179142972103901064656367799729101972135969323384280212697664890134818576566327320339551224766083137358268157135685206321004591730269800124610781727476896167628347329246233920429626004145502166780491946937912046971914841015662929646423256253710776395126958520336218064325

beta(1)
%20 = 5.265557792007748655013092371115806082275788083416767092496807686592646318537349838483082957100807697218634715319914617704079125086410923234779061912897693893915319405948476744902352809614410679494663556344851261085928796868356161949249788828717970604243208924621885557001698203363492134961277837011039856345709922644830790268853421336362576162637971908867335599081278989117876710855805837490367919748669368851066150925882766842582649502470781362709861327653560440689558366042310067084121811240184550799349338313216643491636985749804150282634637249487948908137242809141972073901749714815054656102612403666513896432868358039293215264007471728269804057238755943449106129986465966520551220746527764744415797766770918594388459299772014589292694325536755469770364737152952317211470609361970934307614032515242052181700066886368133325937543598698081176307158189735878908122321654065616007171452444077885812332265243448236602221279147025327081648155219743599331955969230771630055466922331114770725228656394136 E-11681

beta(z+1) - exp(base*beta(z))/(1+exp(-mult*z))
%21 = 0.E-12682 + 0.E-12677*z + 0.E-12673*z^2 + 0.E-12669*z^3 + 0.E-12666*z^4 + 0.E-12662*z^5 + 0.E-12658*z^6 + 0.E-12655*z^7 + 0.E-12652*z^8 + 0.E-12648*z^9 + 0.E-12645*z^10 + 0.E-12642*z^11 + 0.E-12639*z^12 + 0.E-12636*z^13 + 0.E-12632*z^14 + 0.E-12629*z^15 + 0.E-12626*z^16 + 0.E-12623*z^17 + 0.E-12620*z^18 + 0.E-12617*z^19 + 0.E-12614*z^20 + 0.E-12611*z^21 + 0.E-12608*z^22 + 0.E-12605*z^23 + 0.E-12602*z^24 + 0.E-12599*z^25 + 0.E-12596*z^26 + 0.E-12594*z^27 + 0.E-12591*z^28 + 0.E-12588*z^29 + 0.E-12585*z^30 + 0.E-12582*z^31 + 0.E-12580*z^32 + 0.E-12577*z^33 + 0.E-12574*z^34 + 0.E-12571*z^35 + 0.E-12568*z^36 + 0.E-12566*z^37 + 0.E-12563*z^38 + 0.E-12560*z^39 + 0.E-12558*z^40 + 0.E-12555*z^41 + 0.E-12552*z^42 + 0.E-12550*z^43 + 0.E-12547*z^44 + 0.E-12544*z^45 + 0.E-12542*z^46 + 0.E-12539*z^47 + 0.E-12536*z^48 + 0.E-12534*z^49 + 0.E-12531*z^50 + 0.E-12529*z^51 + 0.E-12526*z^52 + 0.E-12524*z^53 + 0.E-12521*z^54 + 0.E-12518*z^55 + 0.E-12516*z^56 + 0.E-12513*z^57 + 0.E-12511*z^58 + 0.E-12508*z^59 + 0.E-12506*z^60 + 0.E-12503*z^61 + 0.E-12501*z^62 + 0.E-12498*z^63 + 0.E-12496*z^64 + 0.E-12493*z^65 + 0.E-12491*z^66 + 0.E-12488*z^67 + 0.E-12486*z^68 + 0.E-12484*z^69 + 0.E-12481*z^70 + 0.E-12479*z^71 + 0.E-12476*z^72 + 0.E-12474*z^73 + 0.E-12471*z^74 + 0.E-12469*z^75 + 0.E-12466*z^76 + 0.E-12464*z^77 + 0.E-12462*z^78 + 0.E-12459*z^79 + 0.E-12457*z^80 + 0.E-12455*z^81 + 0.E-12452*z^82 + 0.E-12450*z^83 + 0.E-12447*z^84 + 0.E-12445*z^85 + 0.E-12443*z^86 + 0.E-12440*z^87 + 0.E-12438*z^88 + 0.E-12436*z^89 + 0.E-12433*z^90 + 0.E-12431*z^91 + 0.E-12429*z^92 + 0.E-12426*z^93 + 0.E-12424*z^94 + 0.E-12422*z^95 + 0.E-12419*z^96 + 0.E-12417*z^97 + 0.E-12415*z^98 + 0.E-12412*z^99 + 0.E-12410*z^100 + 0.E-12408*z^101 + 0.E-12406*z^102 + 0.E-12403*z^103 + 0.E-12401*z^104 + 0.E-12399*z^105 + 0.E-12397*z^106 + 0.E-12394*z^107 + 0.E-12392*z^108 + 0.E-12390*z^109 + 0.E-12387*z^110 + 0.E-12385*z^111 + 0.E-12383*z^112 + 0.E-12381*z^113 + 0.E-12378*z^114 + 0.E-12376*z^115 + 0.E-12374*z^116 +[+++]


It actually works exceptionally well here! We get 12000 digit accuracy--though that's probably because it's in the neighborhood of zero already!!

For fun, I tried a negative value too; so here's \(\exp(1E-5+\pi i) \approx -1\):

Code:
\r beta.gp
init(1,1E-5+Pi*I)
%18 = (6.370542742196280583271196011243055529609616969230416418975693471398271256043809855993588966505844008389430073737333624266272315633971391655020238482289133595693603849790128084325849637962018872778105017781045480944105604404928676704220427083454387083174477194028999771721571824304859910948562893234920920144238260266350548946099572300884919501437655419713569084209637174288915301918557676343391028562876220465491003641203916859245637484901596954001267485931828154549822425601127129480258643144133494835531165837467795638548699283010420808905842684274737267878448077630692084009235093601932309417771074465417168611782862529727540585132533424491509111741165342825341702392648620450351697075130677239674525456173761513139933256484527047373904372381671987450941491160974330261434292971439612825007000742556997028841204139096302316396801379176073556530669543809836490102472143883833429375843678290681961061786627858276086720738123020158264971460586756379130763600803819506362335324386508437435173940198640 E-133 - 8.6549194480179078100250356345375662348390403555734463835315064376278081499758091500079127306832798907291609062661381712675245507350531109736502841315863489658788498409250059265884559523014613982402318213204231080001663718229688196798536016094468032225248382907146440490053534536410115061902555327014878060445767814975103641891965247332011805290470169038006302564869743161505133388888531891975103328661652195878003788082926247129659460462172601793592185341287957734218570913345834989073506021404355393802218199263964905227287527043730098365560759171303032685191198709337109427768800315090286880478269055132089734416302064455191741029679806382276824957409921233674716860952588358701357623946504131292729532641423056357018916165650135801711621547809500122867461275373315997341423076112373305671715081307618825141353738979258738320844623051734445651256480092346950377444886604644471088066558448113134178559381767100696435986739483353591480146473561984685169132214906681535804556646[+++]

beta(0)
%19 = 0.2375809024724632975931570104134484867590810055377534726198893919385751544178949333395340493896414265874058610553215106647052057476004312632239759730305903724988350505822127659060244001810496771526834792898470436413854627348062344741800766541510386799572850273625881766731045316881717781376884077104629297745796850622489298177157696740715924505522201335340104258491145231928135946513019751526285542662796319506791371260538978245943968874823927374650933450397200252003238071313253518071332519529765302726989086439398983628581832897877877535591696974241053053623206330489162050396076961506060441217031638048093151172796857184320175795491643132621937621268903072058647815847074925659299113730152378921450929911232876840002420169225419555271117534123275624337359455074230383313687703613616526240346067470689485904450631352357308928108531908127200480336073422071294722670143902549013070236970724521446507905856675203577220922350710987648100335114125423256916248941252806521725577434198559366687155902338519 + 0.0914015925546030014730606778290236333082793068389504132259911445426104166804664779693079675099388861302950360451592473789813354601984899401430480532468820712158281550841024734533911551593106026551662132277066773050792140318549211747008089221344268485579245727742427214148724359229884663331324871544739201087645372177984182730568027126464807774116974358665260800166032216872940602837146027068197900592897963555197795563699033530234133464068864523956743629614065004651033493059194964840869270710178161617809675441126828150974368954311512911309379295983492259332145803850233923844524569761070331532032193571628199858154702158749263665225922403233114321898310677821409327450750545895589556487992546829874535978667939264319268174663128229577657022340777438373002106866318296584170322357518557153799787221688500953744796687284847624166263013588410913447941141348130426153487798196522337622793446136258974003169534569567235301905633392963605504982103509995380854991596366827880655819017457[+++]

beta(z+1) - exp(base*beta(z))/(1+exp(-mult*z))
%20 = (0.E-1002 + 0.E-1002*I) + (0.E-1002 + 0.E-1002*I)*z + (0.E-1002 + 0.E-1002*I)*z^2 + (0.E-1002 + 0.E-1002*I)*z^3 + (0.E-1002 + 0.E-1003*I)*z^4 + (0.E-1003 + 0.E-1003*I)*z^5 + (0.E-1003 + 0.E-1003*I)*z^6 + (0.E-1004 + 0.E-1003*I)*z^7 + (0.E-1003 + 0.E-1004*I)*z^8 + (0.E-1004 + 0.E-1003*I)*z^9 + (0.E-1003 + 0.E-1004*I)*z^10 + (0.E-1004 + 0.E-1004*I)*z^11 + (0.E-1004 + 0.E-1004*I)*z^12 + (0.E-1004 + 0.E-1005*I)*z^13 + (0.E-1005 + 0.E-1004*I)*z^14 + (0.E-1005 + 0.E-1005*I)*z^15 + (0.E-1005 + 0.E-1005*I)*z^16 + (0.E-1006 + 0.E-1005*I)*z^17 + (0.E-1005 + 0.E-1007*I)*z^18 + (0.E-1006 + 0.E-1006*I)*z^19 + (0.E-1006 + 0.E-1006*I)*z^20 + (0.E-1006 + 0.E-1006*I)*z^21 + (0.E-1007 + 0.E-1006*I)*z^22 + (0.E-1007 + 0.E-1007*I)*z^23 + (0.E-1007 + 0.E-1007*I)*z^24 + (0.E-1007 + 0.E-1007*I)*z^25 + (0.E-1007 + 0.E-1008*I)*z^26 + (0.E-1009 + 0.E-1008*I)*z^27 + (0.E-1008 + 0.E-1008*I)*z^28 + (0.E-1008 + 0.E-1008*I)*z^29 + (0.E-1009 + 0.E-1009*I)*z^30 + (0.E-1009 + 0.E-1011*I)*z^31 + (0.E-1009 + 0.E-1009*I)*z^32 + (0.E-1009 + 0.E-1009*I)*z^33 + (0.E-1010 + 0.E-1010*I)*z^34 + (0.E-1012 + 0.E-1010*I)*z^35 + (0.E-1010 + 0.E-1011*I)*z^36 + (0.E-1011 + 0.E-1011*I)*z^37 + (0.E-1011 + 0.E-1011*I)*z^38 + (0.E-1011 + 0.E-1012*I)*z^39 + (0.E-1012 + 0.E-1011*I)*z^40 + (0.E-1012 + 0.E-1012*I)*z^41 + (0.E-1012 + 0.E-1012*I)*z^42 + (0.E-1013 + 0.E-1012*I)*z^43 + (0.E-1012 + 0.E-1013*I)*z^44 + (0.E-1013 + 0.E-1013*I)*z^45 + (0.E-1014 + 0.E-1013*I)*z^46 + (0.E-1013 + 0.E-1014*I)*z^47 + (0.E-1014 + 0.E-1014*I)*z^48 + (0.E-1014 + 0.E-1014*I)*z^49 + (0.E-1014 + 0.E-1015*I)*z^50 + (0.E-1015 + 0.E-1015*I)*z^51 + (0.E-1015 + 0.E-1015*I)*z^52 + (0.E-1015 + 0.E-1015*I)*z^53 + (0.E-1016 + 0.E-1015*I)*z^54 + (0.E-1016 + 0.E-1016*I)*z^55 + (0.E-1016 + 0.E-1016*I)*z^56 + (0.E-1017 + 0.E-1016*I)*z^57 + (0.E-1016 + 0.E-1017*I)*z^58 + (0.E-1017 + 0.E-1017*I)*z^59 + (0.E-1018 + 0.E-1017*I)*z^60 + (0.E-1017 + 0.E-1019*I)*z^61 + (0.E-1018 + 0.E-1018*I)*z^62 + (0.E-1018 + 0.E-1018*I)*z^63 + (0.E-1018 + 0.E-1019*[+++]


Still good convergence, even at 300 iterations as opposed to a thousand.

For a thousand iterations I also tried \(-1E13\) which is super damn slow; but it does work too:


Code:
\r beta.gp
init(1,log(1E13)+Pi*I,1000)

 *** _/_: Warning: increasing stack size to 16000000.
%18 = (-53.31037925066047495673165268607566441060365320209951830282670512703518311905332670954897290121003340367347357576638427942737937756655593614601607209759121030827503565619841325761556391154604610468510111965458382814985131009223932810190407029070225046430545584912318250790448362245942045049589331809730923269574870470808442360621205470709671079952557682950082377712860589176605123968802154554734000105549388030404295316238939643391885358072412563342930139553870241956852147318635128124613345132482589609663508679636901029548496969223908804339500375971626672237476164305355270359838440298473570362628525868412276194237242950046140483531073862807263539616812016414375959857346896994961152631401213145192600901338040934890407376745125523095576228691417792767330796837166899072370248841196971299551592644226081860927266078051497827472465470925766117713227577958701379764648555273059609762185926282045081233463183367564336086542437082503942918649096826001301885423724496810234803512440043964146286728517128 + 7.360266450946606755625339770333231845245059316767878834707669433349582579698705679375088260495642617530138134733551116609406241839542768326904283548473193366268015016236391417670521530789415025484177693337625047940864134806104301477874738357730591534309267298242545400362573090174920086990930087883201651066035998874856241974667003733583794984431919667503280401579565890660180885110352758965608649621892896628823826387686651376173038190006643295198267568951320772835250914368680685144195847990332390454391664023197210059921009043869594981012419398126609816378029130401222839254538445489315570569158347806383311635394703364574744427001475653431583018626887827292355366900932956410596875556787885032085517908966247663803954384614231433279941473399453185838093758720609524468331781270517531376035978375037197341661166859383409145433415621020965772586879920235580416461369413113234746973780911738844491642489800874273806242745638250827972967975133116570641137577661640280561683165401241[+++]

beta(0)
%19 = 2374786044089840508.270373732865472285161792703072583005034504320194897296670573110837203240279256081603942183319086677183028720843243531229080010474875678012148164186407880707264009247341200758940123967501889134191088819836203819151942790043384826476009729913001256798782503297239819612701552141094212367233331987156005900798669827615011526079851801418585799459663265262065925001088311871574944475351457536161876312855082668765883319410744297287166301817905127131824786913560241493155729102734137130829472327012483773922021894397483083518132348322081855704229075884480068444292825646493944435428204498018092081992912750648714796294317067892154584099765754346599471091643393589821244659627753355167664220649837539570592103625993817252995195926049446520380000176743009232682257740186300752443747231498644777960983344214054213595393039498642586066617782041493170862407846964185575572988554713651335008417734348579556114781395708037760841032713748786400309843845537758122524885122150228658261517857919834 + 4103623123749473973.77728961531082338601381620270223050426065179658586676471486781023594905770724546208806728913012470253528330712440772348740692386065753947275211882773979185553712127352488560584076021548247602435327719315128445422846410838075144506147848173745705489759100070528788111089114078298687234615770483038268330317730680651015174044978417912556449706058880110023542141150933432447054585877303105295951859118461159487083443343032691616075402317101699240702237036638782629937777499675331913132106350591130331295590070392742833578660053346373961233824343131598078636100549664866772994222781709660482396108992013277206912876928659542609210140342933860308136321735405671588789826106094862856327524425212212835254195036686261387010949756552535112210452939802962300058237302499026159029271324620999032863078078327531539208025619546338237108521308011704072267286403155882050199810730426205621810436995668027051933842025940389759417587400088586359661074532932782240888482666250711293[+++]

beta(-1)
%20 = 1.576927162543843683068676760531718814822676338730656516858579136595788492336132800688293070762318508803138981429985818207717876685777173540104705735493304044602143662714041753915901499441871830797655458287381942237208238006028239196034539114521063996187434879456085514663475002603080289341187107347379136418718491622433949286386943489133450506037351512496306100716860422064673398852077728896733104048944816918756931635901605002968459407626434986399542103024780503455698037774609695784019103816792650864238081070496134250435841850684175060109760834843284449877884593909003311744000138384827209182731737328621450698785092722445774104906417562212910767991147633060421070587927783404796862485803332925874247439338578443863571197868749406043782778201644845953260106098970969834226568228710767682072090896343931733216100371833233703348527813026390577088844355856550462352053325103816786875145210473159004521311146503879351667580014866187239592247699557138608304072426314741198659815901269899955587941507821 + 0.91896866246456071337594185222758607751752647337614380119720677653580272914954942654699823210612636646678742823800966153758960585768562198708046128351172038047209266517745169324938203255244932413360323056951736133483210002582399249115831421733469902532555663262519396684115822251219309063861779847348494904983356350175166765104559202284566339520452103462607779906833212116763992784055947994757762915392633079210177899853383809859777735039269561791035667012008461589924735530411927153705866644238669553801206045657130084494295175351029908889739292137063644831833949990685958384429820159378175373542791410364485183699889572233145669015140271491329016767271717181663979721485668418543142789692030876826854801072945975232474794780933726134912039585800265903628995121213980072187330836497064185989816381753518631788889033816066079032476399083457939758032760305694674429287839139838697683600154515082462556644226036616469952855715518373530609971769339341771115734711654580087267671583295840[+++]

beta(z) - exp(base*beta(z-1))/(1+exp(-mult*(z-1)))
%21 = (0.E-983 + 0.E-983*I) + (0.E-981 + 0.E-980*I)*z + (0.E-978 + 0.E-978*I)*z^2 + (0.E-976 + 0.E-976*I)*z^3 + (0.E-974 + 0.E-974*I)*z^4 + (0.E-972 + 0.E-972*I)*z^5 + (0.E-970 + 0.E-971*I)*z^6 + (0.E-969 + 0.E-969*I)*z^7 + (0.E-967 + 0.E-967*I)*z^8 + (0.E-966 + 0.E-965*I)*z^9 + (0.E-964 + 0.E-964*I)*z^10 + (0.E-962 + 0.E-963*I)*z^11 + (0.E-961 + 0.E-961*I)*z^12 + (0.E-960 + 0.E-959*I)*z^13 + (0.E-958 + 0.E-958*I)*z^14 + (0.E-956 + 0.E-957*I)*z^15 + (0.E-955 + 0.E-955*I)*z^16 + (0.E-954 + 0.E-953*I)*z^17 + (0.E-952 + 0.E-952*I)*z^18 + (0.E-951 + 0.E-951*I)*z^19 + (0.E-950 + 0.E-950*I)*z^20 + (0.E-948 + 0.E-948*I)*z^21 + (0.E-948 + 0.E-947*I)*z^22 + (0.E-946 + 0.E-946*I)*z^23 + (0.E-944 + 0.E-945*I)*z^24 + (0.E-943 + 0.E-943*I)*z^25 + (0.E-942 + 0.E-942*I)*z^26 + (0.E-941 + 0.E-941*I)*z^27 + (0.E-939 + 0.E-940*I)*z^28 + (0.E-938 + 0.E-940*I)*z^29 + (0.E-937 + 0.E-937*I)*z^30 + (0.E-936 + 0.E-936*I)*z^31 + (0.E-935 + 0.E-934*I)*z^32 + (0.E-933 + 0.E-934*I)*z^33 + (0.E-932 + 0.E-935*I)*z^34 + (0.E-931 + 0.E-931*I)*z^35 + (0.E-930 + 0.E-930*I)*z^36 + (0.E-929 + 0.E-928*I)*z^37 + (0.E-928 + 0.E-928*I)*z^38 + (0.E-926 + 0.E-927*I)*z^39 + (0.E-925 + 0.E-925*I)*z^40 + (0.E-924 + 0.E-924*I)*z^41 + (0.E-924 + 0.E-923*I)*z^42 + (0.E-922 + 0.E-922*I)*z^43 + (0.E-921 + 0.E-921*I)*z^44 + (0.E-920 + 0.E-920*I)*z^45 + (0.E-919 + 0.E-919*I)*z^46 + (0.E-919 + 0.E-917*I)*z^47 + (0.E-917 + 0.E-916*I)*z^48 + (0.E-915 + 0.E-916*I)*z^49 + (0.E-914 + 0.E-915*I)*z^50 + (0.E-913 + 0.E-913*I)*z^51 + (0.E-913 + 0.E-912*I)*z^52 + (0.E-912 + 0.E-911*I)*z^53 + (0.E-910 + 0.E-910*I)*z^54 + (0.E-909 + 0.E-910*I)*z^55 + (0.E-908 + 0.E-908*I)*z^56 + (0.E-907 + 0.E-907*I)*z^57 + (0.E-907 + 0.E-906*I)*z^58 + (0.E-905 + 0.E-905*I)*z^59 + (0.E-904 + 0.E-904*I)*z^60 + (0.E-903 + 0.E-905*I)*z^61 + (0.E-902 + 0.E-902*I)*z^62 + (0.E-901 + 0.E-901*I)*z^63 + (0.E-901 + 0.E-900*I)*z^64 + (0.E-899 + 0.E-899*I)*z^65 + (0.E-897 + 0.E-898*I)*z^66 + (0.E-897 + 0.E-897*I)*z^67 + (0.E-896 + 0.E-896*I)*z^68 + (0.E[+++]

And it still works; though I suspect 1000 iterations is the key again.




Either way, I don't see the errors you are talking about. But remember I'm running \(\exp(bz)\) as opposed to \(b^z\); whereas my old code ran with the latter. It fixes a good amount of issues.

PS: I think I figured out how to solve the normalization process; it requires renormalizing each step of the way. So that every \(F_n(0) = 1\) including \(F_0(z) = \beta(z+k_0)\). So in reality, we want to run the program \(F_n(z) = \log^{\circ n} \beta(z+n+k_n)\). I'm just trying to find an efficient manner at the moment. This solves the problem of runaway normalization constants for multipliers and bases; where the final normalization constant can move in the thousands by doing slight adjustments to the code.


Regards, James


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - sheldonison - 10/26/2021

Fascinating James,

I look forward to spending more time understanding the \(b=\sqrt{2}\) solutions, and how those solutions interact with the iterating functions "logbase" ... and how the two fixed points, 2 and 4 effect the solution, which has a much better chance of being analytic than base(e).   Time is limited though during the working week.  It would be interesting to see if this solution converges to the Schroeder solution as the logbase period gets larger, where the Schroeder solution is \(\frac{-2\pi i)}{\ln(\ln(2))}\approx17.143i\) periodic.  My hunch is that the beta solution does not ... though how it behaves should be interesting.  Perhaps we will see superexponential growth from the fixed point of four???  Maybe near \(\Im(z)=18i...19i+n\)


RE: A first hires look at tetration \(\lambda = 1\) and \(b = e\) - Ember Edison - 10/26/2021

1. So which is more "analytic", the beta method or Kneser's method? I can't relate Sheldon's discovery to the complex plane

2. And do you have any idea to get beta method Super-logarithm and super-root?

3. I was very surprised by the robustness of the beta method, and I'm still narrowing down the base to see if the beta method crashes completely first, or if Pari-GP crashes first. Does the beta method have a limit when base -> 0? Can remove the singularity 0?