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The recent posts on the Taylor series for the superfunctions of $\eta=exp(1/e)$ reminded me that I want to post my theory, that as imag(z) increases, the lower and upper superfunctions at eta converge towards each other, plus a constant. Here, sexp(z) is the lower superfunction, with sexp(0)=1, and cheta(z) is the upper superfunction, normalized so that cheta(0)=2e.
$\text{sexp}_{\eta}(z)=\text{cheta}(z+\theta(z)+ k )
$
. Where $\theta(z)$ is a 1-cyclic function, which quickly decays to zero as imag(z) increases. Then, the constant "k" is 5.0552093131039000 + 1.0471975511965977*I. And then we have for any real number x,
$\lim_{z \to i\infty}\text{sexp}_{\eta}(x+z)=\text{cheta}(x+z+k)$
- Sheldon
(05/23/2011, 09:01 PM)sheldonison Wrote: [ -> ]Where $\theta(z)$ is a 1-cyclic function, which quickly decays to zero as imag(z) increases. Then, the constant "k" is 5.0552093131039000 + 1.0471975511965977*I.

Without looking at numerical values I at least want to mention, if theta is 1-cyclic and bounded (which would follow from decay towards ioo) then theta would be constant, because every bounded entire function must be constant.

PS: you have some typos in the thread title.
(05/23/2011, 09:49 PM)bo198214 Wrote: [ -> ]
(05/23/2011, 09:01 PM)sheldonison Wrote: [ -> ]Where $\theta(z)$ is a 1-cyclic function, which quickly decays to zero as imag(z) increases. Then, the constant "k" is 5.0552093131039000 + 1.0471975511965977*I.

Without looking at numerical values I at least want to mention, if theta is 1-cyclic and bounded (which would follow from decay towards ioo) then theta would be constant, because every bounded entire function must be constant.

PS: you have some typos in the thread title.
Yeah, if the 1-cyclic function is a sum, $\theta(z)=\sum_{n=1}^{\infty}a_n \exp(2n\pi zi) + b_n \exp(-2n\pi zi)$, then all of the b_n terms must be zero, so it decays to zero as imag(z) increases. The theta(z) function is a Kneser mapping function, which is the sum of the a_n terms, with singularities for integer values of z. It is defined if imag(z)>0, or (if imag(z)=0 and z is not an integer). Then the sexp(z) function for imag(z)<0 is defined by a Schwarz reflection. And, just like sexp base e, the sexp_eta(z) winds up with singularities for integers <= -2, and the singularity cancels for integers>-2.

I was curious to see if anyone else had noticed this. Unfortunately, I don't know how to prove it. edit: In a prevoius post Henryk pointed out that if you have a sickle between the two conjugated fixed points, then that is a required condition. I'll have to try to understand that better ....

I have also numerically calculated the terms for theta(z), and verified that $\text{sexp}_{\eta}(z)=\text{cheta}(z+\theta(z)+ k)$. At imag(z)=I, the upper harmonics have already decayed enough that the graph looks very sinusoidal. Here is the graph of theta(z) at imag(z)=1, from -1+i to 1+i, where the amplitude of the main harmonic has decayed to 0.00017. Closer to the real axis, as I have I previously posted, the graph has has progressively more and more high frequency terms, due to the singularities at integer values of z.
[attachment=851]
- Sheldon
Ah, ok, I see $\theta$ is not entire.

Interesting conjecture. It seems also confirmed by the pictures (made by Dmitrii) on page 8 in the attached "Computation of the Two Regular Super-Exponentials to base exp(1/e)".
(05/24/2011, 01:06 PM)bo198214 Wrote: [ -> ]Ah, ok, I see $\theta$ is not entire.

Interesting conjecture. It seems also confirmed by the pictures (made by Dmitrii) on page 8 in Computation of the Two Regular Super-Exponentials to base exp(1/e).

I look forward to reading Dimitrii's paper. Apparently, it must still be pre-publication, as all I can see is, "You are not the submitter of that submission".

I originally thought I would calculate theta(z) by generating a Kneser Riemann mapping, which would require some modifications to my Kneser.gp code, that I haven't had time to try yet. So yesterday I calculated $\theta(z)$, the easier way, using the equation.
$\theta(z)=\text{cheta}^{[-1]}(\text{sexp}_\eta(z))-z$. With 100 terms, I get double precision accurate results for imag(z)>=0.04, for the equation generating
$\text{sexp}_\eta(z) = \text{cheta(z+\theta(z))$

$\theta(z)=\sum_{n=0}^{\infty}a_n \exp(2n\pi zi)$
- Sheldon
Code:
a0=   5.055209313103899981427271658      + 1.047197551196597746154214461*I a1=   0.001401432408004982064340007511   - 0.08904740918967849870934295177*I a2=   0.0009745731324078295832984646629  - 0.03733654159670666560784931807*I a3=   0.0007404889205790805265231636913  - 0.02245163221999563066686678902*I a4=   0.0005977370286484969934606112522  - 0.01566861320844214271503958136*I a5=   0.0005018188095335350931986608551  - 0.01186529332434380263977231767*I a6=   0.0004328859262727893114218595443  - 0.009460928317981022798940378931*I a7=   0.0003809014595550165678571492537  - 0.007816731171743851556941559870*I a8=   0.0003402618234487611101481798529  - 0.006628163352342624227383260109*I a9=   0.0003075939639173547159799215463  - 0.005732675461843755198639234095*I a10=  0.0002807453713338013423295582876  - 0.005036044996709619239965901548*I a11=  0.0002582771296160210171482485512  - 0.004480100652114545710028919199*I a12=  0.0002391903418640395307331931933  - 0.004027093132427529121265916221*I a13=  0.0002227695461439846829236163650  - 0.003651515085320291655116272034*I a14=  0.0002084886246539934650313328597  - 0.003335544497851734132558987053*I a15=  0.0001959519083684979759009401366  - 0.003066373271924652665766726615*I a16=  0.0001848560084002174716493723357  - 0.002834572553826099457593729541*I a17=  0.0001749643283845140793275542082  - 0.002633055428186743467041405775*I a18=  0.0001660895957555459653575353496  - 0.002456397759533825563676697530*I a19=  0.0001580816120917391360034662861  - 0.002300381255325041020144947863*I a20=  0.0001508184876206487505933950793  - 0.002161678602253742036691677320*I a21=  0.0001442002546949378251879899043  - 0.002037631860904034394761237794*I a22=  0.0001381441386597940781844807782  - 0.001926093525313687540525613089*I a23=  0.0001325810044912043965298881451  - 0.001825310579968987293842353802*I a24=  0.0001274526512991445206423146700  - 0.001733838619440749736308054928*I a25=  0.0001227097273938544436717948304  - 0.001650477347446570539452342914*I a26=  0.0001183101057521097997277481361  - 0.001574221516901694917931394154*I a27=  0.0001142176053280640982610535210  - 0.001504223180398029010010727678*I a28=  0.0001104009751420239103339480127  - 0.001439762333281498046237794053*I a29=  0.0001068330801480615201768928431  - 0.001380223858719170924344196858*I a30=  0.0001034902435606454180893773006  - 0.001325079257143679586322607042*I a31=  0.0001003517116034386350043137768  - 0.001273872045026330000851413737*I a32=  0.00009739921485915265960428031391 - 0.001226205994486849087707605909*I a33=  0.00009461660644770768700046168391 - 0.001181735591722903965021864817*I a34=  0.00009198956175853567401040883751 - 0.001140158242709145552137267935*I a35=  0.00008950532784095349671393699161 - 0.001101207865430173830537972247*I a36=  0.00008715251311607190862114059486 - 0.001064649590333464612069084113*I a37=  0.00008492091002943633794542510503 - 0.001030275352561567534730881761*I a38=  0.00008280134476984375462942275341 - 0.0009979002063759414723269818780*I a39=  0.00008078554934854265757344611947 - 0.0009673592279551277559038828657*I a40=  0.00007886605224633117965160234413 - 0.0009385049002703592238092463243*I a41=  0.00007703608455452840253694241964 - 0.0009112048950695282598856714279*I a42=  0.00007528949910455451712399993731 - 0.0008853401836432737041236171679*I a43=  0.00007362070053381184813176832374 - 0.0008608034211188673268738387616*I a44=  0.00007202458459834668661645557936 - 0.0008374975593584600623149174001*I a45=  0.00007049648533491644198131195480 - 0.0008153346517509591916977910073*I a46=  0.00006903212891155828414424440119 - 0.0007942348197522282270039963471*I a47=  0.00006762759319810549452667164173 - 0.0007741253563048465999970066706*I a48=  0.00006627927224529327986141118065 - 0.0007549399455309304136905173243*I a49=  0.00006498384499013811403086384826 - 0.0007366179815511834322739092355*I a50=  0.00006373824761166039284572731576 - 0.0007191039721045764908375238263*I a51=  0.00006253964904908745268584344639 - 0.0007023470149538662745232624375*I a52=  0.00006138542926786393657800524872 - 0.0006863003369629445325235286598*I a53=  0.00006027315991985189231878357399 - 0.0006709208873018764360001615791*I a54=  0.00005920058709522020318657689901 - 0.0006561689775371129512439945485*I a55=  0.00005816561590646858275542935056 - 0.0006420079624476251587449823567*I a56=  0.00005716629668123243268962093353 - 0.0006284039563124871656274774584*I a57=  0.00005620081257112882568742301639 - 0.0006153255801737527027070039023*I a58=  0.00005526746840987326816593983832 - 0.0006027437362161538543361550203*I a59=  0.00005436468067599215950564266765 - 0.0005906314059431095605653168242*I a60=  0.00005349096843430907824955542661 - 0.0005789634692837646506509988087*I a61=  0.00005264494514651459542980164377 - 0.0005677165421521567773724005657*I a62=  0.00005182531125496916334211020872 - 0.0005568688303084867346171496857*I a63=  0.00005103084745579309692254392628 - 0.0005463999976531811621498207293*I a64=  0.00005026040858756305014135536644 - 0.0005362910473246820762957458455*I a65=  0.00004951291807080781710529778611 - 0.0005265242141780421307604904750*I a66=  0.00004878736284118448539690992239 - 0.0005170828673987320219591051588*I a67=  0.00004808278872589237062774724498 - 0.0005079514221589755815619342667*I a68=  0.00004739829621869267192971986904 - 0.0004991152593560911923917617956*I a69=  0.00004673303661396948706830070118 - 0.0004905606525868100353274921784*I a70=  0.00004608620846469680807673367555 - 0.0004822747016109439782823789160*I a71=  0.00004545705433305464642813545665 - 0.0004742452716442620778929015543*I a72=  0.00004484485780584053326459783235 - 0.0004664609378958392019519183419*I a73=  0.00004424894074981422586561521618 - 0.0004589109348310155800475428621*I a74=  0.00004366866078474811455393153752 - 0.0004515851096987690164901333625*I a75=  0.00004310340895428028781111683652 - 0.0004444738799128717961111574169*I a76=  0.00004255260757672155645069581270 - 0.0004375681939206363139573924993*I a77=  0.00004201570825978641296565767696 - 0.0004308594952321623556175000438*I a78=  0.00004149219006483058717854929964 - 0.0004243396893174826062081029054*I a79=  0.00004098155780761016979549091842 - 0.0004180011131094600618343978837*I a80=  0.00004048334048385136454737949561 - 0.0004118365068772362322703988929*I a81=  0.00003999708980905499247496180616 - 0.0004058389882589056999915363140*I a82=  0.00003952237886297257166771909484 - 0.0004000020282632833847693321790*I a83=  0.00003905880083009562360390763433 - 0.0003943194290694666461596057805*I a84=  0.00003860596782830943780909185078 - 0.0003887853034696618081261505890*I a85=  0.00003816350981858787729537434956 - 0.0003833940558156926198601493311*I a86=  0.00003773107358925655914386089294 - 0.0003781403643429529057196647252*I a87=  0.00003730832180893633042725719543 - 0.0003730191647574956225725833469*I a88=  0.00003689493214280477488141250170 - 0.0003680256349826301166887349831*I a89=  0.00003649059642728702615311408530 - 0.0003631551809709722031325226988*I a90=  0.00003609501989871414341345618353 - 0.0003584034234964835098390760686*I a91=  0.00003570792047187275897277322305 - 0.0003537661858487575579576052061*I a92=  0.00003532902806471807697366646524 - 0.0003492394823587570627518463913*I a93=  0.00003495808396583751167940482314 - 0.0003448195076914650278831505013*I a94=  0.00003459484024153777548068892099 - 0.0003405026268465563013993844869*I a95=  0.00003423905917968713672316721656 - 0.0003362853658132924416581629758*I a96=  0.00003389051276767959483582990300 - 0.0003321644028304493570193946070*I a97=  0.00003354898220210128918480081641 - 0.0003281365602062558333803305399*I a98=  0.00003321425742787372469169907900 - 0.0003241987966570974374284096841*I a99=  0.00003288613670482528160722327206 - 0.0003203482001271648896125235552*I

(05/24/2011, 02:18 PM)sheldonison Wrote: [ -> ]Apparently, it must still be pre-publication, as all I can see is, "You are not the submitter of that submission".

Oh, dear, I uploaded it to the arxiv. But it takes some time there until it is public.
So, I attached the file in the original post. It is currently under review in "Mathematics of Computation".
(05/24/2011, 01:06 PM)bo198214 Wrote: [ -> ]Ah, ok, I see $\theta$ is not entire.

Interesting conjecture. It seems also confirmed by the pictures (made by Dmitrii) on page 8 in the attached "Computation of the Two Regular Super-Exponentials to base exp(1/e)".
Dimitrii noticed the similarity too, page 20. "Outside the positive part of the real axis, F3(z) approaches e at |z|->infinity in a similar way as F1 does."
- Sheldon
i have been aware of this as well.

i always said : " ill consider this later " and thought " it is probably trivial and most regular posters can easily prove this "

apart from the actual proof , i came to conjecture - without the gut to post it nor the effort to check it (srr)- :

almost all real-analytic functions of type
exp(1) > a_n > 1
f(x) = sum a_n^x

with a single (real) fixpoint and fixpoint derivate and second derivate equal to eta's

(f(x) has a unique real inverse)

satisfy the same property as the eta^x mentioned in this thread.

but it sounded to weird and silly ... and i guess there had to be many counterexamples. ( laying doubt on " almost all " )

but what do you think are the conditions for this strange property of upper and lower superfunctions ?
(05/25/2011, 11:04 PM)tommy1729 Wrote: [ -> ]with a single (real) fixpoint and fixpoint derivate and second derivate equal to eta's

(f(x) has a unique real inverse)

satisfy the same property as the eta^x mentioned in this thread.

Ähm, which function are you talking about, eta^x?
with eta^x i mean e^(x/e).

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