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 tetration from alternative fixed point sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 06/28/2010, 11:03 PM (This post was last modified: 06/28/2010, 11:21 PM by sheldonison.) (06/27/2010, 06:02 AM)bo198214 Wrote: (06/21/2010, 04:24 PM)sheldonison Wrote: I can make graphs of the real contour of the superfunction developed from the alternative fixed point, with singularities at 0, 1, e, e^e, e^e^e .... I think we need to go into the details here. If I compute the Abel function, I dont get singularities at 1,e,e^e,... for the secondary fixed point. I use something similar to the formula: $\text{slog}(z)=\log_c(\log_\ast^{[n]}(z)-e[2])-n$ where c is the derivative at the secondary fixed point e[2] and $\log_\ast$ is the branch of the logarithm with imaginary part between $2\pi$ and $4\pi$, which implies that $\log_\ast^{[n]}(z)\to e[2]$.Henryk, I'm using the same equation you are. Instead of "-n", I mulitply by c^n before taking the log_c. This is mathematically equivalent, but it seems to help identify a unique branch point independent of the size of "n". Perhaps I can explain what I was trying to say with a graph of the SuperFunction (developed from the secondary fixed point), showing some contour lines. To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from $+/-\infty$. Then I graph the exponent of that contour line, which is repeated one unit away, with img(z)=0 and real(z) varying from -$\infty$ to 0, and the exponent of that contour line with img(z)=0 and real(z) varying from 0..1. The three contours are disconnected from each other, which I think you were pointing out to begin with. The Reimann mapping will not be analytic at the boundary points. The pattern repeats, as shown by the "alt 3pi*i contour line." Each contour line would have alternates repeating endlessly in the diagonal direction. If you follow a path on the Reimann surface of the inverse superfunction from $-\infty$ to $+\infty$, the path you take will be along a diagonal. But yes, there will be singularities at 0,1,e,e^e, ... but only two of them will be of interest to the Riemann mapping. I don't think I'm explaining it very well, apologies. There are many other paths that can be be graphed in the SuperFunction from the alternate fixed point. I included the 2pi*i contour, and the pi*i contour as well, though they aren't my primary concern. They can also be used to generate repeating contour lines. This graph is in stark contrast to the superfunction from the primary fixed point, where the consecutive exponentiations fit each other like a glove, allowing the Riemann mapping to cancel out the singularity. - Sheldon bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 06/29/2010, 06:53 AM (06/28/2010, 11:03 PM)sheldonison Wrote: Henryk, I'm using the same equation you are. Instead of "-n", I mulitply by c^n before taking the log_c. This is mathematically equivalent, I think it is not "equivalent" in the sense of "equal", because complex logarithm does not satisfy log(zw)=log(z)+log(w). I guess my formula lands on a completely different branch of slog, hm. Well I will try to reproduce your curves, with your formula. And dont think that my understanding is that much deeper, perhaps we indeed can develop a secondary fixed point real analytic slog, if we dont start with a "plain" initial region, but with an overlapping initial region/manifold. I am really interested in this unitude/multitude topic. (For example what happens with Kouznetsov's method if we just plug in the secondary fixed point pair?) sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 07/01/2010, 03:31 PM (This post was last modified: 07/01/2010, 04:10 PM by sheldonison.) (06/29/2010, 06:53 AM)bo198214 Wrote: I think it is not "equivalent" in the sense of "equal", because complex logarithm does not satisfy log(zw)=log(z)+log(w). I guess my formula lands on a completely different branch of slog, hm. Well I will try to reproduce your curves, with your formula. And dont think that my understanding is that much deeper, perhaps we indeed can develop a secondary fixed point real analytic slog, if we dont start with a "plain" initial region, but with an overlapping initial region/manifold. I am really interested in this unitude/multitude topic. (For example what happens with Kouznetsov's method if we just plug in the secondary fixed point pair?) The path from the real axis to the secondary fixed point is much less direct. Kouznetsov's method requires three consecutive exponents, (he uses 0, 1, e), and their vertical (along increasing imaginary) paths to the fixed point. The path to the primary fixed point is fairly direct. But not the path to the secondary fixed point. For example, consider the path from real=ln(0.5), 0.5, and exp(0.5) to the secondary fixed point (via the i*3pi contour). These are the real values near the maximum of the i*3pi contour, (see graph in earlier post). Here is a graph of the first two. The third, exp(0.5) path to the secondary fixed point is already very chaotic, with img varying in the hundreds, so I left it off the graph. Notice how much more direct the path is to the primary fixed point is. Using this path to the primary fixed point as a seed, I would imagine Kouznetsov's method would converge very nicely. But it is difficult to imagine Kouzenetsov's method converging to anything meaningful for the path to the secondary fixed point (especially considering the even more chaotic exp(0.5) path). I also looked at the path to the secondary fixed point from i*pi contour, but that appears to be even less well behaved than the path via the i*3pi contour. - Sheldon bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 07/02/2010, 07:37 AM (07/01/2010, 03:31 PM)sheldonison Wrote: But not the path to the secondary fixed point. For example, consider the path from real=ln(0.5), 0.5, and exp(0.5) to the secondary fixed point (via the i*3pi contour). These are the real values near the maximum of the i*3pi contour, (see graph in earlier post). Here is a graph of the first two. *nods* This is what I mean that we dont have an initial "plain" region (and I guess regardless how we choose the first curve, the second will always overlap the first or itself). In the best case we have some initial manifold (self overlapping region). But thats in the moment out of my sight how to apply Kouznetsov's method to a manifold. PS: Dont be shy to upload pictures to the forum. I consider it a long term archive (though a very unsorted yet), while I dont know how long referenced sites would live. bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 07/21/2010, 03:24 AM (07/01/2010, 03:31 PM)sheldonison Wrote: Kouznetsov's method requires three consecutive exponents, (he uses 0, 1, e), and their vertical (along increasing imaginary) paths to the fixed point. As I think a second time about it, why does he need paths to the fixed point? He integrates along the paths Re(z)=1, Re(z)=-1. He merely forces the value of the superfunction to be the fixed point for imaginary part going to infinity. How the path behaves while going to the fixed point is not essential for his computation, isnt it? sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 07/21/2010, 04:58 PM (07/21/2010, 03:24 AM)bo198214 Wrote: (07/01/2010, 03:31 PM)sheldonison Wrote: Kouznetsov's method requires three consecutive exponents, (he uses 0, 1, e), and their vertical (along increasing imaginary) paths to the fixed point. As I think a second time about it, why does he need paths to the fixed point? He integrates along the paths Re(z)=1, Re(z)=-1. He merely forces the value of the superfunction to be the fixed point for imaginary part going to infinity. How the path behaves while going to the fixed point is not essential for his computation, isnt it?After reading the cauchy computation thread, I think the Cauchy algorithm is sensitive to getting a reasonable initial guess that is close enough to get convergence, and is also sensitive to updating the nodes in in an order that helps guarantee convergence. Are there any links on the forum with Riemann (Knesser's solution) mapping results? I remember Jay posted some results. I've been toying with very simple iterative Riemann mapping, and was able to get some semi-reasonable results. sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 08/13/2010, 03:53 PM (This post was last modified: 08/13/2010, 04:01 PM by sheldonison.) (06/28/2010, 11:03 PM)sheldonison Wrote: ....To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from $+/-\infty$. Then I graph the exponent of that contour line, which is repeated one unit away, with img(z)=0 and real(z) varying from -$\infty$ to 0, and the exponent of that contour line with img(z)=0 and real(z) varying from 0..1. The three contours are disconnected from each other, which I think you were pointing out to begin with. The Reimann mapping will not be analytic at the boundary points. .... - Sheldon The complex periodicity is incorrect in this graph, which effects the "alternative 3*pi*i contour" graph. For the period I incorrectly used: period = 2Pi*i/L The correct value is period = 2Pi*i/(L-2*Pi*I) This is because the correct equation for the periodiicty is 2*Pi*i/ln(L) Because L>2Pi*I, the primary ln(L)=L-2*Pi*I. The correct complex period=1.3769+2.1751*I. I haven't verified the rest of the graph, but otherwise, I'm still using the same equations as I used when I made this graph, and those equations should have given correct values for the other complex contours. By the way, the equations are posted here. I found the problem when I wrote a pari-gp script for the secondary fixed point. With the fix, the alternative contour no longer fits snugly against the primary i=0 contour. I'm still interested in seeing if there's any way to Riemann map the contours back to a well defined real axis, but I still assume that it is not possible. - Sheldon sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 11/18/2011, 08:57 PM (This post was last modified: 11/18/2011, 10:16 PM by sheldonison.) (08/13/2010, 03:53 PM)sheldonison Wrote: (06/28/2010, 11:03 PM)sheldonison Wrote: ....To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from $+/-\infty$....... With the fix, the alternative contours no longer fits snugly... I'm still interested in seeing if there's any way to Riemann map the contours back to a well defined real axis, but I still assume that it is not possible. - SheldonI'm pretty sure it is possible to generate an analytic sexp(z) function from the alternative fixed point after all! Looking at my older posts, I was also very very very close to seeing the solution 15 months ago. The problem is that there is more than one way to unwrap the inverse Schroder function into the complex plane, to generate the complex superfunction. From the best graph I previously posted, the correction for how to to unwrap the inverse Schroder function is to rotate the graph, and shrink it, so that superf(z+1)=exp(super(z)). But it would be better to just start over! Anyway, I have much prettier pictures this time, because I'm using Mike/Andy's complex graph coloring scheme. Let's start with the complex superfunction from the secondary fixed point, L=2.0623 + 7.5886i. Here is a color plot of the complex superfunction. The period of the complex superfunction from the secondary fixed point is approximately 1.3769+2.17514i.     If you notice the light grey contour, you're looking at where the complex superfunction traces out the real number line from roughly -infinity to 4,000,000, or roughly from sexp(-2) to sexp(3), or five periods of z+theta(z). The negative real numbers are graphed in cyan. One more unit to the left of the cyan/grey contour, would be the 3pi i imaginary contour. Notice, that its $3\pi i$ instead of $\pi i$, because for this alternative solution, $\Im(\text{sexp}(-3..-2))=3\pi i$. The grey contour needs to get z+theta(z) mapped, so that this grey line becomes the real axis of the alternative sexp(z). I don't know how to calculate the theta(z) mapping, or the mathematically equivalent Riemann mapping, because this alternative sexp(z) function is not nearly as well behaved as the sexp(z) from the primary fixed point. Notice how quickly the function starts misbehaving as real(z) increases and imag(z) increases, above the grey contour. But in theory, it should be possible to calculate a theta/Riemann mapping, which would generate a 1 to 1 bijection between the superfunction from the secondary fixed point, and the upper half of the complex plane. I have some ideas for how to calculate it, although the existing Kneser.gp algorithm will not converge. For comparison, here is the sexp(z) from the primary fixed point. Notice how nicely it is behaved, especially as imag(z) increases away from the real axis, and the function quickly converges to the primary fixed point! I also included the equivalent grey contour, for the real number line from roughly -infinity to 4,000,000, or sexp(-2) to sexp(3).     Here is the path from log(0.5) to the secondary fixed point. I didn't even try to include all of the path from 0.5 vertical to the fixed point. The path is even more chaotic than in my earlier post, with the rotated graph superfunction graph.     Finally, here is what the alternative sexp(z) graph would probably look like. As I said, I haven't calculated it yet, but this would have the requisite sexp(-3)..sexp(-2)=3pi i contour, where sexp(z) at integer values of z has the "z" term coefficient equal to 0, and the z^2 term coefficient also equal to zero. - Sheldon     sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 11/21/2011, 11:19 PM (This post was last modified: 11/21/2011, 11:56 PM by sheldonison.) (11/18/2011, 08:57 PM)sheldonison Wrote: I'm pretty sure it is possible to generate an analytic sexp(z) function from the alternative fixed point after all....I generated a taylor series and theta mapping, from the secondary fixed point. The complex plot is very pretty, and shows the z^3 pattern around sexp(z=-1). At the real axis, visually the sexp_l2(z) function looks as predicted in the previous post (see approximation graph). This sexp_l2(z) function is also analytic everywhere in the upper and lower half of the complex plane, with singularities at the real axis for integer values of z<=-2. So, this is another different analytic tetration solution for base(e), which meets all of the same requirements as the preferred solution, but obviously looks very different at the real axis, and in the complex plane, since it converges to the secondary fixed point, $L2\approx 2.0623+7.58863i$, as imag(z) increases or as real(z) decreases. The period of the superfunction ~=1.3769+2.1751i $=\frac{2\pi i}{\log(\log(L2))}$. Here, for the log(log(L2)), we use $\log(\log(L2))=L2-2\pi i$. Initially, the mistake I made was to use $\log(L2)=L2$, which is a correct alternative way to unwrap the inverse schroder function to the complex plane to generate the superfunction, but it does not allow the theta(z) mapping to generate sexp(z). Like the solution using the primary fixed point, the limiting value for the lower half of the complex plane is the conjugate of the L2.     Here is the graph, at a radius of 0.5, around z=-1, showing the three loops around the unit circle.     Here is the taylor series, also generated around z=-1. I also generated a z+theta(z) for imag(z)>0.1i. To get the two series to converge towards each other, I had to get a very good seed value, and even then, convergence was very slow, requiring perhaps 50 iterations to get these results. The algorithm I used to generate a seed value was to start with sexp(z) from the primary fixed point, and use $\text{sexp}(z-\sin(\frac{2\pi z} {2\pi}))$. I used that to generate an initial theta(z) mapping, which still required tweaking before I could get convergence. Also, I was only able to use a radius of about 0.7 for the sexp(z) function about z=-1, so I wasn't sure if that would converge or not. To help improve convergence, I needed a better initial seed. So I also had to generate a real valued Fourier transform, $\text{sexp}(z+\theta_2(z)$, where theta2(z) had about a half a dozen terms. Both of these functions were iterated against $\text{superf}_{L2}(z+\theta(z))$. The taylor series below is accurate to about 32 decimal digits, when compared to $\text{superf}_{L2}(z+\theta(z))$ for imag(z)>0.1i. $\text{sexp}_{L2}(z-1)=\sum_{n=0}^{\infty}a_n z^n$. After each iteration generating the sexp_l2 approximation at z=-1, from the superfunction approximation, I forced a0=0, a1=0, and a2=0, which was required for convergence. Code:sexp from 2nd fixed point taylor series, generated around sexp(-1)=0. a0=   0 a1=   0 a2=   0 a3=   6.0525428091015848467396904114867 a4=  -0.60849523747536175511806501610675 a5=  -5.8574541962938452347169436100018 a6=  -5.8154757843204092671165810360382 a7=   2.5347219528124619394630198728768 a8=   6.4668821970795532719465378531827 a9=   4.4871156839279985057981689319166 a10= -2.3500517589865009654200487701366 a11= -4.8774610594145425557598915202919 a12= -3.2684402012509980035269413156892 a13=  1.3654919521689999892599664565609 a14=  2.9823665195110101683443466612379 a15=  2.2661831696822414547275937778816 a16= -0.52913207870690413983768829488530 a17= -1.5156285520432398053435680304692 a18= -1.4572517446695661697863665832605 a19=  0.094286830609106271622763115792468 a20=  0.61110636950519339942198876521363 a21=  0.86271978601510899914905441427631 a22=  0.038974421486596403801894699769519 a23= -0.14905777043601514522125408517461 a24= -0.47896585094427931499121014795675 a25= -0.030389143449596281145270404911561 a26= -0.042825998034817123208188951486868 a27=  0.26235902102501291355107613937971 a28= -0.014143244950329277386417867196816 a29=  0.10059827919484809141333725260695 a30= -0.15454513160875109801594287654923 a31=  0.050006138519215390765363753443150 a32= -0.10520795919021832977856144839921 a33=  0.10624377494623484634694597176958 a34= -0.068484597990327511436726680258115 a35=  0.094944460228304998109629565511086 a36= -0.085693727084796985310665688533151 a37=  0.074122213721890768173136590316211 a38= -0.083843984677348783907640163390188 a39=  0.076235966508003484127877597757711 a40= -0.073069389756946858076727994027333 a41=  0.075222762588118941471666671581810 a42= -0.070612019609800271404236550635959 a43=  0.069441350373969278684007195642263 a44= -0.068881006958076546205688133213681 a45=  0.066217164743985940967056670962088 a46= -0.065261808225731542796012464732002 a47=  0.064014547545150315863062824154253 a48= -0.062316195718567494332301728926411 a49=  0.061294332906226487840725743704704 a50= -0.060029039396985886662256396924140 a51=  0.058764049550308815974907383174684 a52= -0.057730167455123192197602214419251 a53=  0.056599488372003113915391804142194 a54= -0.055541057151639765193298395532515 a55=  0.054560157576525914731163221375545 a56= -0.053565536674136590005525205437557 a57=  0.052629716197772950980480888304674 a58= -0.051728553326544954734152425491973 a59=  0.050844493606782859378117101810769 a60= -0.050000501525925342307229440392680 a61=  0.049181281229267069678508024865434 a62= -0.048386033785257460166130157341700 a63=  0.047619527381412221049137805806908 a64= -0.046875076515792168847026835482228 a65=  0.046153557896303522805721778538128 a66= -0.045454760295123160291375736294242 a67=  0.044776065510160199491799413861579 a68= -0.044117595385215902555503712705649 a69=  0.043478330251192648438922293764535 a70= -0.042857106284648360469480988317676 a71=  0.042253521301300292210596172100789 a72= -0.041666683066062752366849813544716 a73=  0.041095876145594636984485302734057 a74= -0.040540545496565613945077962499150 a75=  0.040000002117742705825994214770159 a76= -0.039473680162673857504905441992248 a77=  0.038961041511940238193398136688527 a78= -0.038461538061213742686998410078651 a79=  0.037974682757072796139145638913056 a80= -0.037500000856492225977549714336736 a81=  0.037037036646204616366175962080078 a82= -0.036585365813093889607615969483647 a83=  0.036144578520061409527864163769823 a84= -0.035714285553913653372657407963149 a85=  0.035294117694663812812247714829208 a86= -0.034883720959274005451397406041107 a87=  0.034482758574956584392754574654857 a88= -0.034090909117630668844739474066681 a89=  0.033707865165393714267654270592895 a90= -0.033333333324472619467885396896417 a91=  0.032967032976007199844301758708358 a92= -0.032608695648281905690683339130091 a93=  0.032258064515612316681201977767890 a94= -0.031914893619135479632053443899399 a95=  0.031578947366827529439356158688524 a96= -0.031250000000463649984311504584008 a97=  0.030927835051828540795825673426539 a98= -0.030612244897519756304248381285685 a99=  0.030303030303287137214584066663769 a100= -0.029999999999966282031229404541557 a101=  0.029702970296949837283873331539097 a102= -0.029411764705965021657699041094067 a103=  0.029126213592195908278373798109699 a104= -0.028846153846150793259645857652719 a105=  0.028571428571446783989204643005906 a106= -0.028301886792438526629856969692978 a107=  0.028037383177574696251348668866060 a108= -0.027777777777779823427720230977851 a109=  0.027522935779812846337183948157237 a110= -0.027272727272729551533867006685424 a111=  0.027027027027026594871373241617209 a112= -0.026785714285713708587427338763126 a113=  0.026548672566372367355487980292010 a114= -0.026315789473683846658285369595985 a115=  0.026086956521739115588647915103517 a116= -0.025862068965517422422639852899428 a117=  0.025641025641025545847388991708705 a118= -0.025423728813559251424682876256784 a119=  0.025210084033613403390708306581287 a120= -0.025000000000000210634068453080186 a121=  0.024793388429752111676536615771926 a122= -0.024590163934425784207269337708914 a123=  0.024390243902438972919951753608139 a124= -0.024193548387097878008656721979599 a125=  0.023999999999999874830826012878455 a126= -0.023809523809522177198866059223151 a127=  0.023622047244095016547447625736730 a128= -0.023437500000004730528259615801748 a129=  0.023255813953486943095952593420534 a130= -0.023076923076917084925513343342322 a131=  0.022900763358783736467796311748204 a132= -0.022727272727291509099320846247737 a133=  0.022556390977433940347123100851247 a134= -0.022388059701470332273740288257628 a135=  0.022222222222254205500382914734533 a136= -0.022058823529479415282418061328820 a137=  0.021897810218922467761077074543046 a138= -0.021739130434698909152627982450815 a139=  0.021582733813118207956767051755867 a140= -0.021428571428785024559533003066287 a141=  0.021276595744385680497554700219427 a142= -0.021126760563064445336698085322633 a143=  0.020979020979823863900839059151042 a144= -0.020833333333871461238610179305475 a145=  0.020689655170932338970321143814915 a146= -0.020547945204338179052061514895723 a147=  0.020408163268868439831447630713248 a148= -0.020270270270944964825128555194145 a149=  0.020134228180832249842369921195487 a150= -0.019999999996382145768944457088909 a151=  0.019867549683848916030390073901517 a152= -0.019736842101884200464284895313679 a153=  0.019607843104935080898558359139368 a154= -0.019480519472747153686365539175781 a155=  0.019354838769987628622500042979499 a156= -0.019230769194904427325515899160343 a157=  0.019108280114821735333622468050053 a158= -0.018987341782410682797657178061368 a159=  0.018867924762827561642329485236602 a160= -0.018749999782550921947274006260191 a161=  0.018633539801386849975790975198512 a162= -0.018518518784548856555456574041866 a163=  0.018404908859539817637569621034230 a164= -0.018292681833246517878352101357825 a165=  0.018181816011532742375200430960923 a166= -0.018072291311787620165426068307671 a167=  0.017964075079476270315483234073891 a168= -0.017857137826157011051469312107843 a169=  0.017751471802095868856191038213042 a170= -0.017647072166987844489621007896549 a171=  0.017543870851218341724085525290953 a172= -0.017441838139502430481295643756101 a173=  0.017341018294974304424367295663764 a174= -0.017241450709916318715211642676403 a175=  0.017142892764765992718387963486856 a176= -0.017045355278105614225544052063673 a177=  0.016949106072439699278655991418933 a178= -0.016854276424282969507424937927475 a179=  0.016759868130023045438921363043654 a180= -0.016666214315116921424792262275298 a181=  0.016574604276500489383923106468323 a182= -0.016485031090073264715644393566068 a183=  0.016393526409814549014622248699611 a184= -0.016302232081226604371621173580053 a185=  0.016217205012887436111175962990360 a186= -0.016135166958570918826264050175108 a187=  0.016041578337246276076860565681463 a188= -0.015947465741735751400414051023663 a189=  0.015880811347614127000659372647187 a190= -0.015812268974115238817285732324294 a191=  0.015693934950496055261552846318831 a192= -0.015578762616167368077787740783347 a193=  0.015590629166910915436453027039959 a194= -0.015540807357621775424779779754165 a195=  0.015292819039848609300335564328156 a196= -0.015101732862617970858553288354058 a197=  0.015474986702580039507486900124995 a198= -0.015381046865367596879509632876324 a199=  0.014517304156196916116826663108209 sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 12/05/2011, 11:22 AM (11/21/2011, 11:19 PM)sheldonison Wrote: I generated a taylor series and theta mapping, from the secondary fixed point....Below, there are graphs of the sexp(z) from the secondary fixed point, at the real axis, from sexp(-1.5) to sexp(1.5). I also graphed the first and second derivatives, and the equivalent functions from the primary fixed point. Notice how the derivative goes to zero at integer values of z.     I was able to get fairly clean convergence using two different algorithms, both with identical results. The simplest algorithm, with the quickest convergence required an initialization, very similar to the initialization used in the my kneser.gp program, followed by an initial approximation $\text{sexp_{l2}}(z)=\text{sexp}(z-sin(2z\pi)/(2\pi))$. The initial sexp(z) need only have three terms in its Taylor series. Then this initial approximation required an additional 42 iterations, generating a theta(z) approximation from the secondary fixed point, followed by an sexp(z) approximation, from both the theta(z) and the sexp(z) approximation around z=-1. This gave results accurate to ~32 decimal digits. At each iteration, I forced the first three terms in the Taylor series to zero. - Sheldon « Next Oldest | Next Newest »

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