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 theta and the Riemann mapping sheldonison Long Time Fellow Posts: 676 Threads: 24 Joined: Oct 2008 09/21/2011, 05:23 AM (This post was last modified: 09/21/2011, 10:54 AM by sheldonison.) The equation to convert the theta(z) unit disk to Kneser's Riemann mapping unit disk is as follows, where u(z) is $\theta$ wrapped around the unit disk. $\text{RiemannMapping}(z)=z \times \exp(u(z) \times 2\pi i)$ I will derive this equation, but first some background. A year or two ago, I came up with the idea to generate sexp(z) from the superfunction(z), via the equation: $\text{sexp}(z)=\text{superfunction}(z+\theta(z))$, where the superfunction is developed from the inverse Schroeder function of exp(z). My assumption at that time, is that Kneser's construction also involved theta(z), which isn't exactly correct. I tried to understand Henryk's post on Kneser's construction, as well as Jay Daniel's post and graphs on the Kneser construction. However, my predisposition for believing Kneser's construction involved a Riemann mapping of theta(z), along with my limited background probably combined to prevent me from fully understanding Kneser's construction. My lack of understanding didn't stop me from deriving my own algorithm to iteratively generate Kneser's sexp(z) solution from $\theta(z)$, which turned out to be a big improvement in numerically calculating the sexp(z) function. Showing how the $\theta(z)$ function is connected to the Riemann mapping helps put my algorithm on firmer theoretical grounds. Kneser iterated logarithms of the upper half of the complex plane, via the Schroeder equation, to generate the Chi-Star. The limit equation for the Schroeder function is for exp(z) is: $S(z) = (\lim_{n\to\infty} (\log^{[n]}(z+L)-L)\times\exp(n L)$ where the Schroeder equation here is developed centered around the fixed point, $L\approx 0.318 + i1.337$. I was also iterating logarithms, but the difference is I was iterating logarithms to generate the inverse of the superfunction, and the superfunction and to calculate theta(z). The inverse of the superfunction, can be developed from the Schroeder equation as follows. $\text{superfunction}^{-1}(z) = \log(S(z-L))/L$ The limit equation to generate the inverse schroder equation is $S^{-1}(z) = \lim_{n\to\infty} \exp^{[n]}(L+z\times\exp(-n L))$ If you have the inverse Schroeder equation, then $S^{-1}(L z) = \exp(S^{-1}(z))$. The inverse Schroder equation can trivially be turned into the complex valued superfunction, which has superfunction(z+1)=exp(superfunction(z)). $\text{superfunction}(z) = S^{-1} (\exp(z L))$ As noted earlier, there is a simple exact mathematical connection between theta(z) and the Riemann mapping unit circle in Kneser Riemann mapping. Start by representing theta(z) as a Fourier series, and then equivalently, as a unit disk. $\theta(z)=\sum_{n=0}^{\infty}a_n\times \e^{(2\pi n i z)}$ Equivalently, one can wrap $\theta(z)$ around a unit disk, with the transformation $y=\exp(2\pi i z)$. Let us call this unit disk version of theta(z), "u(y)", where u stands for the unit disk. It is simply theta(z) wrapped around a unit circle, which is a little bit confusing since the theta(z) unit disk is not the same as the Riemann mapping unit disk in Kneser's construction. It does have some similarities, in that u(0) corresponds to $+\Im \infty$, since the theta(z) function was defined so that only positive values of n are used in the a_n terms of the fourier series. The u(y) function uses exactly the same a_n coefficients as the $\theta(z)$ function, but it is a taylor series instead of a fourier series. $u(y)=\sum_{n=0}^{\infty}a_n\times y^n$ The u(y) function is analytic in the unit disk, except for a singularity at u(1), which corresponds to the singularities in $\theta(z)$ for integer values of z. As stated earlier, the equation to convert the theta(z) unit disk to the Riemann mapping unit disk is: $\text{RiemannMapping}(y) = y \times \exp(u(y) \times 2\pi i)$ To derive this equation, one can start with the solution for sexp(z), and use that to generate $f(z)=z+\theta(z)$. $\text{sexp}(z)=\text{superfunction}(z+\theta(z))$ $f(z)= z+\theta(z)= \text{superfunction}^{-1}(\text{sexp}(z))$ f(z+1)=f(z)+1 by definition, since both the superfunction(z) and sexp(z) are superfunctions of exp(z). f(z) cannot be wrapped around the unit circle directly. But according to Jay's post, Kneser's trick is to multiply f(z) by $2\pi i$ and then exponentiate. call this function g(z). $g(z)=\exp(f(z)\times 2\pi i)=\exp((z+\theta(z))\times 2\pi i)$ Now, g(z+1)=g(z), since $\exp(2\pi i)=1$. So we have succeeded in creating a 1-cyclic function g(z). At the real axis, a unit length of g(z) can be wrapped a unit circle. So g(z), starting from z=0, to z=1, can be wrapped around a unit circle, with the interior uniquely defined by the Riemann mapping. A unit length of g(z) can be calculated from [0..1], by starting with any approximation, and then doing a Riemann mapping. Next, recall that $f(z)= z+\theta(z)$, so $g(z)=\exp((z+\theta(z))\times 2\pi i)=\exp(2z\pi i) \times \exp(\theta(z)\times 2\pi i)$ Next, recall the substitution connecting the u(y) function, which is exactly the substition necessary to wrap g(z) around the unit circle $y= \exp(2\pi i z)$ and $z = \frac {log(y)}{2\pi i}$ $\text{RiemanMapping}(y)=g( \frac {log(y)}{2\pi i})$, which is a Riemann mapping of a unit length of g(z), wrapped around the unit disk. And finally, with trivial algebra, $g(\frac {\log(y)}{2\pi i}) = y \times \exp(u(y)\times 2\pi i)$, which derives the formula for Kneser's Riemann mapping in terms of the unit circle of theta(z). $\text{RiemanMapping}(y) = y \times \exp(u(y)\times 2\pi i)$ This can also be reversed, to derive the unit circle theta(z) function from the Riemann mapping. $u(y) = \frac{\log (\text{RiemanMapping}(y)/y)}{2\pi i}$ The Riemann mapping uniqueness criteria required is RiemannMapping(0)=0. Otherwise dividing by y creates a singularity at the origin. The first derivative of the Riemann mapping taylor series at the origin is equal to the $\exp(2a_0\pi i)$ of the a0 taylor series term of the unit circle function, or the theta(z) function. The a0 term shows how in the limit, as imag(z) goes to infinity sexp(z) converges exponentially towards superfunction(z+a0). So, in summary, Kneser's Riemann mapping construction corresponds exactly equivalent to the theta(z) equations developed in my kneser.gp iterative generation of the sexp(z) function. Both the unit circle version of theta(z) and the Riemann mapping unit circle are defined everywhere in the unit circle, except for the singularity at z=1. Using these equations, I have succeeded in recreating all of the graphs in Jay's post, as well as some new ones of my own which I will post later. - Sheldon Levenstein sheldonison Long Time Fellow Posts: 676 Threads: 24 Joined: Oct 2008 09/28/2011, 01:39 PM (This post was last modified: 10/15/2011, 05:50 AM by sheldonison.) (09/21/2011, 05:23 AM)sheldonison Wrote: The equation to convert the theta(z) unit disk to Kneser's Riemann mapping unit disk is as follows, where u(z) is $\theta$ wrapped around the unit disk. .... $\text{RiemanMapping}(y) = y \times \exp(u(y)\times 2\pi i)$ This can also be reversed, to derive the unit circle theta(z) function from the Riemann mapping. $u(y) = \frac{\log (\text{RiemanMapping}(y)/y)}{2\pi i}$ .... Both the unit circle version of theta(z) and the Riemann mapping unit circle are defined everywhere in the unit circle, except for the singularity at z=1. Using these equations, I have succeeded in recreating all of the graphs in Jay's post, as well as some new ones of my own which I will post later. - Sheldon LevensteinFrom my point of view, the biggest obstacle to using the theta(z) in my own program to generate numerical results for sexp(z) is the really nasty singularity that theta(z) has at integer values of z. This is probably also the biggest obstacle to using the Riemann mapping in Kneser's construction to generate numerical results for sexp(z). Both have fascinating very complex singularities, as you superexponentially approach integer values for sexp(z). This is the contour for theta(z)+z, going through 5 iterations, from z=-2 to z=3 $\text{superf}^{-1}(\text{sexp}(z))$, where we approach within sexp(3.5) or 1.6E-78 of the integer values of z.     This is the corresponding Riemann unit circle mapping, approaching within 1.6E-78 of the singularity. Surprisingly, At this level, there is still a visible discontinuity in the Riemann mapping unit circle! Later in this post, I post the numerical values for 100 terms of the Taylor series of the Kneser Riemann mapping unit circle function.     Going back to the theta(z)+z contour, now I fill in the detail, getting superexponentially closer and closer to the singularity at sexp(z=-1)=0. The earlier plot above approached to within sexp(z)=1.6E-78. In green, the first extension goes 1/sexp(3.5) to 1/sexp(4.5). For comparison, a googolplex is roughly sexp(4.53). But the detail increases as we superexponentially approach zero. The next extension, in red goes from 1/sexp(4.5) to 1/sexp(5.5). Sexp(5.5) is a number too larger to meaningfully describe. The next extension, in green, goes from 1/sexp(5.5) to 1/sexp(6.5). And the next extension, in red goes from 1/sexp(6.5) to 1/sexp(7.5). And finally, in green 1/sexp(7.5) to 1/sexp(8.5). I used superexponential approximations to calculate these plots, which was actually quite a difficult calculation! Underneath, is the exact corresponding theta(z) plot. Notice that theta(z) function will slowly approach +real infinity, and -imag infinity, as the function superexponentially approaches the singularity at zero. Also, notice that the singularity becomes ever more and more complicated, winding and unwinding along repeated paths superexponentially close to paths that it has already followed.     For refeence, this is the chi-star contour plot, through 5 iterations, from z=-2 to z=3 $\text{Schroeder}(\text{sexp}(z))$, where we approach within sexp(3.5) or 1.6E-78 of the integer values of z.     I'm not an expert on Riemann mappings, and I've never used any of the kernals that can be used to calculate a Riemann mapping. My own approach is to avoid the singularity, where the functions are more well behaved. For my kneser.gp program, I iteratively calculated the theta(z) mapping at imag(z)=0.12i, instead of at the real axis. This is far enough away from the singularity to avoid all of the numerical difficulties. So, I thought to myself, why not use the exact same idea to iteratively calculate Kneser's Riemann mapping? So, I modified a version of my "kneser.gp" program, to use the Riemann mapping in place of the unit circle theta function, using the equations from the first post in this thread, for sexp(z) base "e". I used the code I developed in kneser.gp, to iteratively generated the Riemann mapping approximations and sexp(z) approximations from each other, using the Riemann mapping unit circle function to give increasingly more accurate approximations for sexp(z) for imag(z)>=0.12i. After fifteen iterations, the result, was a 110 term taylor series for the Riemann mapping unit circle, accurate to 32 decimal digits, for imag(z)>0.12i, along with a taylor series for sexp(z), centered at z=0, and also accurate to 32 decimal digits. This compares with thirteen iterations required for the theta(z) mapping, so this Riemann mapping approach appears to be a little bit less efficient than the theta(z) approach. Here is the resulting Riemann mapping unit circle taylor series that I iteratively generated. The absolute value of the Taylor series terms is slowly decreasing until the 98th term, so the terms beyond that have no numerical value, and are doubling due to random noise in the numerical calculations, since 0.12i corresponds to a radius of 0.47 for the unit circle function. Now that I know definitively that theta(z) is a completely different function than the Riemann mapping unit circle, I will update the naming conventions used in the next version of my kneser.gp program update, to eliminate any references to the Riemann mapping unit circle. I realize that anyone familiar with Kneser's approach must have been confused by my naming conventions. - Sheldon Levenstein Code:a0=   0 a1=  -327.79007374992963622452244710336 - 23.902116636626409517157394798245i a2=  -185.94568152036261113859941661981 - 21.533727429280802462636912155451i a3=  -130.50213757402687605649648347030 - 18.403984947427146004993299195969i a4=  -100.75545181262672532249873197964 - 16.011659628906052417390988193659i a5=  -82.145594696058311070948040454018 - 14.192510029588507698399427523107i a6=  -69.382936867583043925835782956869 - 12.771427099758044545429407317314i a7=  -60.076146970284106210396597105759 - 11.631011880337075726744114004412i a8=  -52.984037950635645046972710891255 - 10.694561459224654541607974304768i a9=  -47.397457322893760854438578178177 - 9.9107563234651959814651056693485i a10= -42.881439436381137469457173570825 - 9.2441782378580933378571333472734i a11= -39.154195512724542330789304842159 - 8.6696463663053873334873340660045i a12= -36.025051669643689817004120471877 - 8.1687763529311455190893636809793i a13= -33.360327416038227243857649908354 - 7.7278293384141073153870303642114i a14= -31.063493736780052760582647014858 - 7.3363265548072539216255751613370i a15= -29.063088712643849264297151099397 - 6.9861314820967664680442554364063i a16= -27.305066277238994291249082958683 - 6.6708259102267120521666125895662i a17= -25.747788787504291396984343530034 - 6.3852758324871054906487986066249i a18= -24.358654910067780958371585738063 - 6.1253230460967895099161899493649i a19= -23.111771492088469815957503628422 - 5.8875619073964711812362801614627i a20= -21.986310583411263697135304877001 - 5.6691749711958499151659500266966i a21= -20.965327209815775886301799689573 - 5.4678101163893734394728667397608i a22= -20.034893781649422274479213013806 - 5.2814874011833510859166519726491i a23= -19.183456362234624361802542624426 - 5.1085275554898491353473971827965i a24= -18.401349128671405250386295708599 - 4.9474964453735940909165573316768i a25= -17.680423426838872845392206297141 - 4.7971614817685213368012861271603i a26= -17.013761041471644942070247395176 - 4.6564570686895579241119094565734i a27= -16.395450175006981815350269344603 - 4.5244569683520423402098714590307i a28= -15.820408687806484588050464796250 - 4.4003520132161439468394716411490i a29= -15.284243355488934859356661378006 - 4.2834319905616506013672229555474i a30= -14.783136857451417486175196327538 - 4.1730708118719968718711894070750i a31= -14.313756320906959052899170266877 - 4.0687142894325671838746435140627i a32= -13.873178768820082822485791011168 - 3.9698699982123392626859187786330i a33= -13.458829933548945059845058971029 - 3.8760988175670444470933290975883i a34= -13.068433720198288136624660889587 - 3.7870078352597721006141097769930i a35= -12.699970216927352886692646865285 - 3.7022443633033773942083689796719i a36= -12.351640611144531414618916624817 - 3.6214908665985295792532743099418i a37= -12.021837721171897221752258095355 - 3.5444606451819840305749015046465i a38= -11.709121121486188386724379190628 - 3.4708941419647317538167593494817i a39= -11.412196046875666901280732711202 - 3.4005557722295620737514649674806i a40= -11.129895421958761792818008933049 - 3.3332311904318800293372265582862i a41= -10.861164488626287745496419320764 - 3.2687249251737034856326294656246i a42= -10.605047603341814655301351673563 - 3.2068583254790576691499767390446i a43= -10.360676855022264495387109367745 - 3.1474677713583344655036404976304i a44= -10.127262217055885835060861835345 - 3.0904031096212031612588052330241i a45= -9.9040829974082989095323066631582 - 3.0355262823762547913159109150591i a46= -9.6904803913983401433552547350763 - 2.9827101209459079176330750272765i a47= -9.4858509746516521859670022202320 - 2.9318372822650365989440657808654i a48= -9.2896410005517061890424289766326 - 2.8827993084077127121081553589574i a49= -9.1013413884416799599125508069072 - 2.8354957928452825976254090123318i a50= -8.9204833068531640467873418020511 - 2.7898336394970834764077875542204i a51= -8.7466342709091076479147986063656 - 2.7457264026850788912781116198564i a52= -8.5793946853694046238358145211779 - 2.7030936978196453187178720353152i a53= -8.4183947750350996473748798323494 - 2.6618606740852066764104203223546i a54= -8.2632918527820305908142569554323 - 2.6219575416094338475432717273759i a55= -8.1137678826619891823873020036605 - 2.5833191466272398257571715451642i a56= -7.9695273015348572484254683483441 - 2.5458845890224895274261596108909i a57= -7.8302950677773801388307525831956 - 2.5095968773720908570697174630777i a58= -7.6958149099144560803530508384218 - 2.4744026172501562741685731978295i a59= -7.5658477516687169944671503908005 - 2.4402517290916964751335541822628i a60= -7.4401702930308590913435182646648 - 2.4070971923802283470181602209108i a61= -7.3185737296058317415347730068344 - 2.3748948133237429220453943580367i a62= -7.2008625947595553974994051597898 - 2.3436030135285237940883612675573i a63= -7.0868537110411326894384085871270 - 2.3131826374788046888853358157767i a64= -6.9763752390289142149002148723068 - 2.2835967768887375288372168760754i a65= -6.8692658132020843132562286590829 - 2.2548106102182031104352115241532i a66= -6.7653737556760313504395153515370 - 2.2267912558388601560706471859024i a67= -6.6645563597524004996218191427609 - 2.1995076375088610338608271860372i a68= -6.5666792361176745646707178111338 - 2.1729303609604138709235118726018i a69= -6.4716157154636473451664511733011 - 2.1470316005403539636332611466148i a70= -6.3792463017750393376527320053851 - 2.1217849949439347009117472920987i a71= -6.2894581716563992322561457644349 - 2.0971655512054801635861434133526i a72= -6.2021447146093804154402023972707 - 2.0731495561404836280216963065747i a73= -6.1172051117495994016993315216235 - 2.0497144946160884571732597124575i a74= -6.0345439466823409635671103549480 - 2.0268389738803043570179900142719i a75= -5.9540708513614922561819673793689 - 2.0045026537310827121078300163267i a76= -5.8757001725994286242166003316126 - 1.9826861815768090401192728320184i a77= -5.7993506813623154476457129552272 - 1.9613711337300757413696118773088i a78= -5.7249452741210282180701571185053 - 1.9405399613800510170867826011348i a79= -5.6524107649590677620027787465303 - 1.9201759494004042711846002734013i a80= -5.5816775614286128580684907406263 - 1.9002631840138964432361981035827i a81= -5.5126796311901175158608242411470 - 1.8807865639729691325050578073284i a82= -5.4453538906228117361507462463368 - 1.8617318333899477030942191189225i a83= -5.3796406727745852553220650143510 - 1.8430857559886439562133683285603i a84= -5.3154816441114804277369870446172 - 1.8248362717227121490960834859722i a85= -5.2528219575520924076272493414467 - 1.8069729592568200850823344978205i a86= -5.1916017115269917512928778034828 - 1.7894867736533517936175099887947i a87= -5.1317648595423870184752913732253 - 1.7723696339490268008127141880373i a88= -5.0732253916353023132867628883406 - 1.7556082869409887369810490624957i a89= -5.0159087616720307523387066374746 - 1.7391727195679999411675277971628i a90= -4.9596304055794727024782499446655 - 1.7229740917686007415929232934887i a91= -4.9043068366648240828733519598773 - 1.7067932333515502941404098691773i a92= -4.8495670107053927698498110924749 - 1.6900885363791222499865386397544i a93= -4.7958216611483793583184653557062 - 1.6717285271408746689997731883507i a94= -4.7431378295283860201554612786050 - 1.6494036958276063887565451303043i a95= -4.6962522478448480349071321779598 - 1.6192002957487002955207240497814i a96= -4.6612346811848425631301574519034 - 1.5750417988645529951157717645377i a97= -4.6663379523926262649080743481408 - 1.5112946157952910916894942179764i a98= -4.7486085069223752235241199733712 - 1.4299728365388148877216760501027i a99= -5.0277925698254062067642361113893 - 1.3695464125059109100976810964625i a100= -5.6301110707162123526948619335240 - 1.4663526553029164208897556113097i a101= -6.8952449085803341405909041212052 - 2.1164760540238249197081274569076i a102= -8.9152704158399421307669489416876 - 4.2684044009680905266185439631413i a103= -11.887152459919960734806857483470 - 10.037048931545010600340464050593i a104= -13.639208049980242193037867553837 - 23.558856388596261932308906608031i a105= -9.2451969879860023889537863913708 - 52.248818470995411127457051445723i a106=  20.002167649568569747079146036915 - 106.97464138549194075325600185014i a107=  112.27833849222522299147525038401 - 198.96173799102239221370038867625i a108=  363.19903721073536593550095166035 - 323.25443713559541377204765592827i a109=  945.99666317573730972320340967517 - 411.13177402160055414902246549332i sheldonison Long Time Fellow Posts: 676 Threads: 24 Joined: Oct 2008 10/11/2011, 12:49 PM (This post was last modified: 10/14/2011, 07:31 PM by sheldonison.) I've been playing around with Mike's pari-gp implementation of the HSB graphing system, and I thought I'd add a busy plot, showing the evolution of the superfunction (on top), to the sexp(z) function (on the bottom), with the z+theta(z) contour highlighted in yellow. I repeated the contour between the two graphs, with more details. The superfunction is exactly Periodic, with a period=4.4470 + 1.0579i; below the sexp(z) function converges to the same period as imag(z) increases. I'm using the HSB graphing system unmodified, however, I used some tricks to preserve as much accurate information as possible, when iterating exp(z) to huge numbers, while eventually treat the imaginary component as a random number, to model the chaos as the points randomly go from huge positive, to huge negative. So this plot, in addition to showing the superfunction, and the sexp(z), also shows one unit length of the z+theta(z) mapping linking the two functions. The highlighted yellow path shows one unit length, developed by taking the inverse superfunction from z=0 to z=1. You can also see the the cyan contour from the previous iteration, going from -infinity to 0. In the HSB graphing system, negative numbers are shown in cyan, and positive numbers are shown in red, with zero being black. The larger the magnitude, the brighter the plot. For larger magnitudes approaching 1E100, the saturation increases to pure white. The rainbow in the sexp(z) function occurs at approximately sexp(2.5 to 3), as the sexp(z) function begins to swings from large positive to large negative numbers. I tried to show how z+theta(z) switches back and forth, turning around infinite number of times as necessary, as it takes an infinitely long path from the inverse_superfunction(0) to the inverse_superfunction(1). I cut off the infinite section going towards superfunction(z)=1. The yellow contour gets mapped to sexp(-1) to sexp(0). If the yellow contour is repeated, then everything above the yellow contour has a 1 to 1 bijection with the upper half of the complex plane for sexp(z). Sometime, I will post the pseudo recursive relationship, governing the pseudo repeating pattern, and the turn arounds in the contour, which occur at 1/sexp(n) for integer values of n>=5; the graph in the middle shows some of the pattern. - Sheldon « Next Oldest | Next Newest »

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