• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Infinite tetration of the imaginary unit sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 06/21/2011, 03:02 PM (This post was last modified: 06/21/2011, 09:24 PM by sheldonison.) (06/21/2011, 12:13 PM)Gottfried Wrote: Waahh... I must have had a period of really weak thinking. Welcome to the club.... Quote: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*Iedit 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 « Next Oldest | Next Newest »

 Messages In This Thread Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 09:52 AM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 11:35 AM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 01:28 PM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 08:53 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 09:10 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/14/2008, 08:25 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/15/2008, 05:35 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/19/2011, 09:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 05:27 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:20 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 01:36 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 02:46 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 12:13 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 03:02 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 08:00 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/22/2011, 09:28 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/22/2011, 03:23 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/23/2011, 08:55 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/23/2011, 01:13 PM solved -- they're called Siegel discs - by sheldonison - 06/23/2011, 06:10 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:11 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/20/2011, 10:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 01:58 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/21/2011, 11:19 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 03:15 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 07:05 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 08:07 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/24/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/24/2011, 04:36 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 09:44 PM RE: Infinite tetration of the imaginary unit - by bo198214 - 06/26/2011, 08:06 AM

 Possibly Related Threads... Thread Author Replies Views Last Post [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 1,527 03/20/2018, 12:16 AM Last Post: tommy1729 [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 13,876 12/28/2014, 10:22 PM Last Post: MphLee Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 5,958 05/06/2014, 09:47 PM Last Post: tommy1729 Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 7,916 07/17/2013, 09:46 AM Last Post: Gottfried Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,808 09/06/2012, 02:01 AM Last Post: JmsNxn The imaginary tetration unit? ssroot of -1 JmsNxn 2 6,004 07/15/2011, 05:12 PM Last Post: JmsNxn the infinite operator, is there any research into this? JmsNxn 2 6,075 07/15/2011, 02:23 AM Last Post: JmsNxn Tetration and imaginary numbers. robo37 2 6,035 07/13/2011, 03:25 PM Last Post: robo37 Infinite Pentation (and x-srt-x) andydude 20 29,309 05/31/2011, 10:29 PM Last Post: bo198214 Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) Gottfried 91 103,364 03/03/2011, 03:16 PM Last Post: Gottfried

Users browsing this thread: 1 Guest(s)