Infinite tetration of the imaginary unit

[sheldonison]

Waahh... I must have had a period of really weak thinking.

Welcome to the club....

Well, this discussion brought me back to the right track, I suppose. After I got it now (again, I must have had it already earlier) I made a picture, how the fixpoint, the log of the fixpoint, and the base according to the Shell-Thron-description are connected....In my notation I always used u (for the log of the fixpoint), t =exp(u) for the fixpoint and b=exp(u/t) for the base....

Thanks Gottfried, great picture. So, using your notation, we add one more equation: \( \text{period}=2\pi i/\log(u) \). In the original example, with all of the nice plots you made, base=1.712936040374417981826i.
u = -0.5070842426299714169803 + 0.8618964966145228964379i
period = 2.988300627934001489933

Now, if you take one of those plots you made, starting with one of your initial points, you'll find that the period works exactly. For example, if you start iterating with b^^0 = 0.7+0.7i. then
\( \text{Superfunction}(0)=0.7+0.7i \)
\( \text{Superfunction}(x \bmod \text{period})=\exp_b^{o x}(0.7+0.7i) \)
And if you repeat this an infinite number of times, then you get the nice periodic contour curve you made. I'm not sure about what the correct notation is for mod with an irrational number. The SuperFunction is real periodic, where iterating (x mod 2.9883) times gets you back to your starting point.
\( \text{SuperFunction}_{L}(x) = L + \sum_{n=1}^{\infty} a_n\exp(2n\pi ix/\text{period}) \)
Using the exact same coefficients, the circular contour of the function you graphed can be represented as f(z), evaluated on the unit circle, where abs(z)=1.
\( f(z) = L + \sum_{n=1}^{\infty} a_n z^n \)

Here are the 100 coefficients for these equations, which is accurate to 15 decimal digits or so. a0=L. The equations were generated with b^^0=0.7+0.7i
- Sheldon

Code:

a0=   0.3920635484599088334159 + 0.4571543197696414316818*I
a1=   0.1926347048245026996218 + 0.2553626064604919992564*I
a2=   0.04367562340340935922713 + 0.02205560982867233582908*I
a3=   0.007530714487061478885377 - 0.004281576514011399313377*I
a4=   0.04709123530117242894461 + 0.001729440845159316720314*I
a5=   0.01306847282348073176814 - 0.005728356074317823223406*I
a6=   0.0006614004482465856090804 - 0.003745545557771203176425*I
a7=   0.009018437522021720431834 - 0.01057073314544308526135*I
a8=   0.001367493374159604282404 - 0.005020786666321506753021*I
a9=  -0.001093179701013968437283 - 0.001263132425285266404038*I
a10= -0.0009612888490325652001380 - 0.004685330087439725794279*I
a11= -0.001197476243914925206686 - 0.001632842246574827208293*I
a12= -0.0007333426764609148904379 + 0.00002704910834568717554014*I
a13= -0.001585607063510229010734 - 0.0007749689289491325438067*I
a14= -0.0008122799208911508244806 - 0.00002786102261464546068704*I
a15= -0.0001912817298718253167176 + 0.0002595615603574707578201*I
a16= -0.0006062049020599662754753 + 0.0003026807801206704776212*I
a17= -0.0002133612201397169986658 + 0.0002546714235074630134220*I
a18=  0.00003743767676364786366434 + 0.0001366516889627626748407*I
a19= -0.00004960231979796590150008 + 0.0002622624914391191734664*I
a20=  0.00002914410828772657407477 + 0.0001343295228982834413090*I
a21=  0.00005740673450802333073318 + 0.00002412575934410344993297*I
a22=  0.00007190414851207875090348 + 0.00007932051866760387225803*I
a23=  0.00005181266163292071488059 + 0.00002460336753897918612847*I
a24=  0.00002388309209425730931165 - 0.00001336900104575625063695*I
a25=  0.00004334222830648700442252 - 0.000003957863248238146555857*I
a26=  0.00002144406955373366943567 - 0.00001097237534884560319531*I
a27=  0.000001798338756957510733360 - 0.00001189640167775918700159*I
a28=  0.000009381044976804181706355 - 0.00001521809568526962885314*I
a29=  0.000001814217743399466423905 - 0.00001000368045094924883848*I
a30= -0.000003653404747697099145868 - 0.000003824946827096487902828*I
a31= -0.000002722276064735807862073 - 0.000006885441806388226474380*I
a32= -0.000002908243010370422712642 - 0.000003178655514749099491204*I
a33= -0.000002312499568376790395264 + 0.0000002460508774499087597595*I
a34= -0.000002980300482220947987573 - 0.0000008054002380194513555136*I
a35= -0.000001823918641511708691392 + 0.0000001690025918878855565754*I
a36= -0.0000005334119278536951372042 + 0.0000008723944367755511233820*I
a37= -0.000001016497503905268380668 + 0.0000008015125616246374896681*I
a38= -0.0000004112254538462517744914 + 0.0000006638895287291174927516*I
a39=  0.0000001620819496611418767756 + 0.0000004194202258747815345800*I
a40=  0.00000001618007685693369358734 + 0.0000005451229963514326146522*I
a41=  0.0000001209168813396943102704 + 0.0000003110475973992931378252*I
a42=  0.0000001898717133881552152764 + 0.00000005522749966037761739770*I
a43=  0.0000001894442243720459740995 + 0.0000001316472729147078143581*I
a44=  0.0000001376003918043586154671 + 0.00000003856956234095862926094*I
a45=  0.00000006989793638646620143736 - 0.00000005174316419871478195931*I
a46=  0.00000009272637028549195160717 - 0.00000003149903997060054611874*I
a47=  0.00000004862361433330243915387 - 0.00000003733152323435119037092*I
a48=  0.0000000004594161970077067830909 - 0.00000003824731911971521292835*I
a49=  0.00000001262869999583407154392 - 0.00000003974147931865195611863*I
a50= -0.0000000003496984553054855241969 - 0.00000002633575753285570565953*I
a51= -0.00000001329027566459892886883 - 0.00000001031672759687675209111*I
a52= -0.00000001047177328908874356543 - 0.00000001439926459715597781841*I
a53= -0.000000009176173196441562162962 - 0.000000006649128397591598125552*I
a54= -0.000000007145022906860095624727 + 0.000000001929181197324110824839*I
a55= -0.000000007620140099414333325288 - 3.665361956240423108017 E-11*I
a56= -0.000000004657112126987005058983 + 0.000000001463721624742275648158*I
a57= -0.000000001214340042927719948489 + 0.000000003020802059448936620889*I
a58= -0.000000001945865674907896473438 + 0.000000002626112862490583268957*I
a59= -0.0000000006895093416586537172232 + 0.000000001989304633944696536150*I
a60=  0.0000000007408669730258221080258 + 0.000000001225840780840576126459*I
a61=  0.0000000004162667478872884401106 + 0.000000001341526002164161883378*I
a62=  0.0000000005132292039301019798648 + 0.0000000007502246684352801693042*I
a63=  0.0000000006270046072871590876051 + 6.314128778383910279034 E-11*I
a64=  0.0000000005719223588070705693565 + 1.963669989355466718298 E-10*I
a65=  0.0000000003925541915150384351025 + 1.423200907055391980274 E-11*I
a66=  1.876665420232897808715 E-10 - 2.041099568060325883782 E-10*I
a67=  2.138545944166912423998 E-10 - 1.491616932082952807018 E-10*I
a68=  1.057114697606634964409 E-10 - 1.330106086409156802997 E-10*I
a69= -2.245771490160356215933 E-11 - 1.199192549169948489927 E-10*I
a70=  2.412064012682073922868 E-12 - 1.125620287250565717984 E-10*I
a71= -1.925135162098333576978 E-11 - 7.098313998094113622532 E-11*I
a72= -4.819517928618243903621 E-11 - 2.365349936790643533816 E-11*I
a73= -3.871371903062367023679 E-11 - 2.951358803573412657219 E-11*I
a74= -2.971987439383358491286 E-11 - 1.148400477824120990266 E-11*I
a75= -2.102115223518953224343 E-11 + 1.078339509214737316490 E-11*I
a76= -2.019396751549899274826 E-11 + 6.026259356931675413551 E-12*I
a77= -1.163396908506564200969 E-11 + 7.445550617031170199083 E-12*I
a78= -1.716154888316130092496 E-12 + 1.025476040032673922023 E-11*I
a79= -3.036635954368037678981 E-12 + 8.613530463400133051522 E-12*I
a80= -3.752862064975309936775 E-13 + 5.980193219987777616114 E-12*I
a81=  3.172166960395143993046 E-12 + 3.298059914357276464051 E-12*I
a82=  2.248284621116987511317 E-12 + 3.262575232338994828013 E-12*I
a83=  1.999543732722426455567 E-12 + 1.663406533225183694885 E-12*I
a84=  1.997678205175370980314 E-12 - 2.563711028673351106773 E-13*I
a85=  1.720034892688777377316 E-12 + 4.239470389705304190096 E-14*I
a86=  1.095569514705621826147 E-12 - 2.755807745754857768578 E-13*I
a87=  4.327405733493612596926 E-13 - 7.737101605156168079589 E-13*I
a88=  4.538806176142139212252 E-13 - 5.942948240331469394435 E-13*I
a89=  1.836221482061442061104 E-13 - 4.553107307228522798696 E-13*I
a90= -1.605165844309677626103 E-13 - 3.556796760694313558042 E-13*I
a91= -9.344945167389884815526 E-14 - 3.115596977506452337354 E-13*I
a92= -1.130977149566327841894 E-13 - 1.810443673536964897811 E-13*I
a93= -1.679204373998674274001 E-13 - 3.619539889304637767143 E-14*I
a94= -1.337060593554487395913 E-13 - 4.659807725069687201088 E-14*I
a95= -9.263704449807770513808 E-14 - 6.497426500663588784562 E-15*I
a96= -5.670632585997269746189 E-14 + 5.002869038220030918858 E-14*I
a97= -5.058776182826300737460 E-14 + 3.528909585714982918781 E-14*I
a98= -2.596942498401231015211 E-14 + 3.097159125854603958149 E-14*I
a99=  2.925281234940225845834 E-15 + 3.316299012434386946739 E-14*I
a100 = 2.925281234940225845834 E-15 + 3.316299012434386946739 E-14*I

edit By the way, these coefficients also work for other initial values, such as superfunction(x-0.2364561612687660028188 + 0.4934489697206941114376i), for b^^0 = 0.5+0.5i, which corresponds to abs(z)=0.3543311933142207352068.

And all of this seems to work great as long as the period is an irrational real number. But, as you noticed, from the example with a real period = exactly 3, iterating three times doesn't get you back to your starting point....

Finally, the fractal singularity comes from log(0), when you have b^^0=1, then b^^-2= singularity; but then b^^n is also a singularity for all negative integers <=-2, which means, since the function is real periodic, with an irrational real period, that there are singularities everywhere, when b^^0=1. For the coefficients above, the fractal behavior appears to be at around abs(z)=1.298, or around imag(x)=-0.124i.
- Sheldon