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eta as branchpoint of tetrational
#34
Continuing the discussion from my previous post, where we consider the behavior of the superfunction along a circular arc, from base= to base=. For a complex base at the Shell-Thron Boundary, how does the superfunction evolve as we circle closely around base=? And also, can we develop a merged Fatou like mapping from the two fixed points at that base?
Code:
results for a complex base at the Shell-Thron boundary
base  = 1.4749205843145635852862061112 + 0.0034981478091858060388588917267i  
L1    = 2.4372341100745143830388282062 + 0.82552153847929520132720783617i (circular fixed point)
|log(L1)|=1.0000000000000000000000000000 (circular)
Period1=  18.886173137606161572331556476 (real period)
L2    = 2.5206834347439820199663539286 - 0.87677210299493726158947155475i (repelling fixed point at this base)
This L1 fixed point superfunction at the Shell-Thron boundary, with a real period=18.89 is very interesting. My calculations match the mathematical model of the superfunction from L1 being a periodic function, with a real period of approximately 18.89, which decays to L1 as . I generated the Fourier series for the function, and I also generated a plot of the sexp(z) function for this base. To generate the sexp(z) plot, I started with z=1, and iterating z=B^z, for 16000 iterations, subtracting iterations of the real Period. The pattern repeats with a real period of 18.89 to +/- real infinity. Notice that sexp(0)=1, sexp(-1)=0, and notice the approximation of the singularity at z=-2, where we would have log(0). My assumption is that none of the derivatives of the function here are defined anywhere in this cross section, but that for imag(z)>singularity(z), the function is analytic. As the radius approaches closer and closer to eta, this function's real period goes to infinity, and this function becomes the lower superfunction for base eta.
   


Here are the first 80 theta(z) Fourier series terms, which is accurate to 32 decimal digits for imag(z)>=1.5i. This Fourier series for the superfunction, was developed with the somewhat arbitrary requirement that the a_1 term=1. As , the superfunction decays to a periodic function with a period of ~18.89, circling L1 with an exponentially shrinking radius. At imag(z)=-i, which appears to be just slightly above the sexp(z) fractal singularity plot, the superfunction approximation is still reasonably accurate, to +/- 0.1 for most points, and approximates the sexp(z) plot above.
Code:
a1=   1
a2=  -0.0936201434689961368086647459633896 - 0.579238923230953573245678436083305*I
a3=  -0.320683212371393341813787362082673 + 0.126743588529914531522707124081828*I
a4=   0.119616793864212915772264614599621 + 0.164007639553429767557332132175408*I
a5=   0.0732389126380630280505692013728790 - 0.0946974356166440404102912124319164*I
a6=  -0.0663742380831858892066723849627545 - 0.0244308418378105429681555493459044*I
a7=  -0.00113917177691053516060857128440486 + 0.0418461132213518228505660201678252*I
a8=   0.0235941882293188633555260249973232 - 0.00770820240348467892917041034129283*I
a9=  -0.00920933800613021623902496158444828 - 0.0115141419178171674893277574536941*I
a10= -0.00437869312492190154118567427835918 + 0.00760287408841178916234910646887856*I
a11=  0.00518243461543642777367630749190422 + 0.000703172062265366709132895289567522*I
a12= -0.000822335040668600992619235143434256 - 0.00301247557659202033418543708431202*I
a13= -0.00145021915279076526248432300509339 + 0.00117634654885793149100658597647531*I
a14=  0.00100173652080411074133996349716316 + 0.000496505353409563045900142392722522*I
a15=  0.0000106106561384327391968341507990486 - 0.000669811132053360353384377725305466*I
a16= -0.000362051071335304189862533507630806 + 0.000173006216111239867118005085654564*I
a17=  0.000191999420945188253150066666075303 + 0.000143514630594805499331330577520006*I
a18=  0.0000180105057854429502171089437309786 - 0.000140692046300459536323603068136406*I
a19= -0.0000722817144413949442010819146488065 + 0.0000353580254861441148050161200387193*I
a20=  0.0000623843250976460079037027858671792 + 0.0000622994206876068031507183802208773*I
a21=  0.0000394144279898223425421990281491789 - 0.0000644399032387350978471352209618663*I
a22= -0.0000552399546784824284020360655198103 - 0.0000173717300761573123065860634846381*I
a23= -0.00000170900299301120308864944637450145 + 0.0000417901226616108337504224748852074*I
a24=  0.0000282825602470756133396057950556508 - 0.00000700465114159728374550033785950976*I
a25= -0.0000102741251642115188421455104766420 - 0.0000169513861018562623392242118417184*I
a26= -0.00000861450276185261307846582779272648 + 0.0000100872080731087760276251983926557*I
a27=  0.00000817321818082630500200232724077194 + 0.00000318851816155030353724964963717565*I
a28=  0.000000128226838107579182941357184499617 - 0.00000575088366455444677949785380908752*I
a29= -0.00000353092927538484212483341617268819 + 0.00000125499228716754783926743818230882*I
a30=  0.00000160003688794248510453112332724949 + 0.00000182794597930009973297955250749624*I
a31=  0.000000697557688269020001469882035409570 - 0.00000140542576055879895912891047819070*I
a32= -0.00000100889900661331713626426196439158 - 0.0000000564465426616676164278197787755211*I
a33=  0.000000230704122240189039616621944108059 + 0.000000606898414029889666096531558352536*I
a34=  0.000000290650131894774291241762981855498 - 0.000000298068834574435782124277390943904*I
a35= -0.000000252796710132249881563394657044474 - 0.0000000838064588282311556318206599351208*I
a36=  0.0000000262525023567587387196047136774841 + 0.000000167679864301056914917435645077030*I
a37=  0.0000000843156787032266336785583097242438 - 0.0000000661239762044488524011014511173683*I
a38= -0.0000000617506417564549303284536829588438 - 0.0000000200216896839990662943508218014508*I
a39=  0.00000000435332496521962851123056565020908 + 0.0000000571323648825386598048750467464987*I
a40=  0.0000000429334793323671365061159929480907 - 0.0000000166404992504174499695999020022156*I
a41= -0.0000000209040338191777245365860609109214 - 0.0000000275465438797457343240840060614878*I
a42= -0.0000000148019299709549500031744130735256 + 0.0000000200070426208339939250862704370370*I
a43=  0.0000000164127232147400360617236888494594 + 0.00000000583426861205331198249205244563447*I
a44=  0.000000000405211095766879927886426992315563 - 0.0000000119547707093560428329006970617555*I
a45= -0.00000000777496117032379804962737474521166 + 0.00000000228864503143700362724336343241042*I
a46=  0.00000000315506457738290530801554495546432 + 0.00000000443047184583984061051276451431495*I
a47=  0.00000000206832355615669591012934756235558 - 0.00000000297689533811372157097801754876968*I
a48= -0.00000000232884755383860191534522210742407 - 0.000000000595568786314963087888383006153234*I
a49=  1.87943022217748645613168888408468 E-10 + 0.00000000157724751299808970393836445878456*I
a50=  0.000000000918930069471866341849704968939179 - 0.000000000501880286639411066538240515028660*I
a51= -0.000000000537196806513684729782755840381852 - 0.000000000431608328308169134705317212956109*I
a52= -1.20478708183463435222238448963980 E-10 + 0.000000000436782054534548353758126518996639*I
a53=  0.000000000294236873163964332731548860943212 - 4.55731004522102747603632604275855 E-11*I
a54= -1.09484348258184241455411404313358 E-10 - 1.62030719351912555496652240574897 E-10*I
a55= -6.29394314688713941314843553884342 E-11 + 1.11742766424676835413587432779026 E-10*I
a56=  8.36255959878877945937704291986788 E-11 + 1.10291562562161012484293879712436 E-12*I
a57= -2.90513568874658489576233017176322 E-11 - 4.39013600472770972673122513274529 E-11*I
a58= -2.29811153532266632771543167176271 E-11 + 3.63101011822347872513939835257048 E-11*I
a59=  3.25899687388650229010887282083843 E-11 + 7.74390022108389232195175114220041 E-12*I
a60= -1.81868183907415228508371813317456 E-12 - 2.47987652017774192289218548315937 E-11*I
a61= -1.65748690876379100905877610466545 E-11 + 6.56094294458485238945177789424690 E-12*I
a62=  7.89671428527872708294832099697662 E-12 + 9.65558082056524898977076618765452 E-12*I
a63=  4.62125774179362854874547306781964 E-12 - 7.22876756098381830518401046879771 E-12*I
a64= -5.65759620988165606459426764197572 E-12 - 1.40447397757024384653801223882907 E-12*I
a65=  3.59230043053171442330785946914775 E-13 + 3.90929031426616411142571951897310 E-12*I
a66=  2.38110114449910913864737055760666 E-12 - 1.11019536640239763020515574170534 E-12*I
a67= -1.24446898293292890148742241252767 E-12 - 1.23110283238676648386146968644129 E-12*I
a68= -4.69370002626149629187539175418096 E-13 + 1.06415995463525241744580231150137 E-12*I
a69=  7.70901530570370360423583613971827 E-13 + 3.11212443861388366484406050529848 E-14*I
a70= -1.73362371646043940990070178815011 E-13 - 4.81224041607396039360455113531847 E-13*I
a71= -2.49580112873293178112472146938238 E-13 + 2.29709331425666375587118836029636 E-13*I
a72=  2.06048029752544915803470623281604 E-13 + 9.07772793408009314804167345432811 E-14*I
a73= -1.99981988952511503900017626279692 E-15 - 1.49659766697857426926842845765267 E-13*I
a74= -8.91032922459351328104300575456857 E-14 + 4.45637690786538844143278688005410 E-14*I
a75=  5.38920627291828173069098689493713 E-14 + 3.85014408140980596462765810389930 E-14*I
a76=  8.39672845827494984430565262413313 E-16 - 4.28745581469085423500119831314564 E-14*I
a77= -3.07833860695434498416121080170439 E-14 + 1.39064038624055069395977971825958 E-14*I
a78=  1.82651480046632644960792775815767 E-14 + 1.84286959022402725882644514360893 E-14*I
a79=  8.56674454279163246970950744512832 E-15 - 1.71479132326068121784553996632081 E-14*I
a80= -1.35903799211189751496637159522325 E-14 - 1.97835940687435308751982365458644 E-15*I

Reply


Messages In This Thread
eta as branchpoint of tetrational - by mike3 - 06/02/2011, 01:55 AM
RE: eta as branchpoint of tetrational - by mike3 - 06/03/2011, 10:57 PM
RE: eta as branchpoint of tetrational - by mike3 - 06/04/2011, 09:08 AM
RE: eta as branchpoint of tetrational - by mike3 - 06/04/2011, 09:50 AM

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