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 How it looks (i.θ)ₐ sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 04/19/2015, 11:19 PM (This post was last modified: 04/19/2015, 11:43 PM by sheldonison.) (04/19/2015, 08:40 PM)marraco Wrote: Do you think that $^xa=a^{^{x-1}a}$ is still valid on the complex plane? Did you checked its accuracy on the real line?Yes to both. The pari-gp algorithm I wrote will generate a ~33 decimal digits accurate solution in a few seconds, for base(e). The algorithm generates a Taylor series at the real axis, as well as a 1-cyclic $\theta(z)$ mapping for the complex valued Koenig's solution. Both solutions are equal, but the Koenig's theta mapping can be used anywhere in the upper half of the complex plane, except right near the real axis, where an arbitrarily large number of terms in the $\theta(z)$ mapping would be required due to the $\theta(z)$ singularity for integer values of z. $\text{sexp}(z) = \text{Koenig}(z+\theta(z))$. The 109 term Taylor series approximation for $\text{sexp}(z)=\sum_{n=0}^{\infty}a_n\cdot z^n\;\;$ would be accurate to ~33 decimal digits within a radius<=1; the radius of convergence of the infinite series is 2. By definition, $\text{Koenig}(z+1) = \exp(\text{Koenig}(z))\;$ and is entire for repelling fixed points, so if $\theta(z)=\sum_{n=0}^{\infty}a_n\cdot \exp(z \cdot 2\pi i n)$ then $\theta(z+1)=\theta(z)$ and therefore $\text{Koenig}(z+1+\theta(z+1)) = \exp(\text{Koenig}(z+\theta(z)))\;\;\;\$ and if $\text{sexp}(z)=\text{Koenig}(z+\theta(z))\;$, then $\text{sexp}(z+1)=\exp(\text{sexp}(z))$. For ~33 decimal digits of accuracy, the 1-cyclic approximation would require 94 terms if $\Im(z)>=0.12i$. Also, $\theta(z)$ goes to a constant as $\lim_{z \to i \infty}$. There is also a complex conjugate version of the $\theta(z)$ mapping for the Im(z)<0 half of the complex plane. More iterations requires more precision, so it runs a lot slower. Generating such a >70 digit accuracy solution requires 28 iterations and takes 40 seconds. My $\theta(z)$ mapping is mathematically equivalent to Kneser's Riemann mapping. Code:{sexp= 1 +x^ 1*  1.0917673512583209918013845500271517 +x^ 2*  0.27148321290169459533170668362354901 +x^ 3*  0.21245324817625628430896763774094856 +x^ 4*  0.069540376139987373728674232707469711 +x^ 5*  0.044291952090473304406440344385514804 +x^ 6*  0.014736742096389391152096286915534111 +x^ 7*  0.0086687818172252603663803925296399219 +x^ 8*  0.0027964793983854596948259913011495569 +x^ 9*  0.0016106312905842720721626451640260303 +x^10*  0.00048992723148437733469866722583243492 +x^11*  0.00028818107115404581134526404129643988 +x^12*  0.000080094612538543333444273583009977844 +x^13*  0.000050291141793805403694590114624236685 +x^14*  0.000012183790344900091616191711098613934 +x^15*  0.0000086655336673815746852458045541327926 +x^16*  0.0000016877823193175389917890093176351289 +x^17*  0.0000014932532485734925810665044317554713 +x^18*  0.00000019876076420492745531981897955236756 +x^19*  0.00000026086735600432637316458216085893667 +x^20*  0.000000014709954142541901861412188201900914 +x^21*  0.000000046834497327413506255093709947352846 +x^22* -0.0000000015492416655467695218054651322677986 +x^23*  0.0000000087415107813509359129925581627587247 +x^24* -0.0000000011257873101030623175751344777492435 +x^25*  0.0000000017079592672707284125656087948039197 +x^26* -0.00000000037785831549229851764921435415800623 +x^27*  0.00000000034957787651102163178731457999981781 +x^28* -1.0537701234450015066294258497294163 E-10 +x^29*  7.4590971476075052807322860078252226 E-11 +x^30* -2.7175982065777348693298760969650549 E-11 +x^31*  1.6460766106614471303885109354654427 E-11 +x^32* -6.7418731524050529991474333392275879 E-12 +x^33*  3.7253287233194685443170989538379133 E-12 +x^34* -1.6390873267935902234582225010471847 E-12 +x^35*  8.5836383113585680604885670142883911 E-13 +x^36* -3.9437387391053843135799065799345925 E-13 +x^37*  2.0025231280218870558933488201935127 E-13 +x^38* -9.4419622429240650237203369464349681 E-14 +x^39*  4.7120547458493713408134162135725801 E-14 +x^40* -2.2562918820355970800481578125058583 E-14 +x^41*  1.1154688506165369962907704017701283 E-14 +x^42* -5.3907455570163504918533616427279775 E-15 +x^43*  2.6521584915166818728179847001994855 E-15 +x^44* -1.2889107655445536819394124602816863 E-15 +x^45*  6.3266785019566604528387812745585687 E-16 +x^46* -3.0854571504923359890162184894469925 E-16 +x^47*  1.5131767717827405270960446817215775 E-16 +x^48* -7.3965341370947514333418316897203391 E-17 +x^49*  3.6269876710541876035237240096591600 E-17 +x^50* -1.7757255986762984029994538367761954 E-17 +x^51*  8.7098795443960546502631507727982294 E-18 +x^52* -4.2692892823391563141718062049000602 E-18 +x^53*  2.0950441625755281093483690055067076 E-18 +x^54* -1.0278837092822587892363235029846949 E-18 +x^55*  5.0468242474381763889822749445669773 E-19 +x^56* -2.4780505958215521454269990958399214 E-19 +x^57*  1.2173942030393317020113788775364947 E-19 +x^58* -5.9816486323037815150562410082210709 E-20 +x^59*  2.9402643445138969080986317689280122 E-20 +x^60* -1.4455835436201850220023758753334795 E-20 +x^61*  7.1095903736074123276947876938966522 E-21 +x^62* -3.4974025487872032482054204148733769 E-21 +x^63*  1.7209641671884893968788892238552706 E-21 +x^64* -8.4703003156215453164497615979164144 E-22 +x^65*  4.1700185394242509196286233654876760 E-22 +x^66* -2.0534096527658659704995748057858764 E-22 +x^67*  1.0113838615494863919451981634503253 E-22 +x^68* -4.9825424396849504025363950795013587 E-23 +x^69*  2.4551694611351628954393221798701698 E-23 +x^70* -1.2100465472324105299447061203120237 E-23 +x^71*  5.9650225602500368204601428440073432 E-24 +x^72* -2.9410859968904307308168910746569497 E-24 +x^73*  1.4503990988357537983984682685714610 E-24 +x^74* -7.1539937380082747218197703756972717 E-25 +x^75*  3.5293042057319186187079609197431644 E-25 +x^76* -1.7414327791202733830945057246997563 E-25 +x^77*  8.5940845951550628378204439831675622 E-26 +x^78* -4.2419517528160810175697295824342703 E-26 +x^79*  2.0941281695727407230153052186977015 E-26 +x^80* -1.0339757529559312470973226618639183 E-26 +x^81*  5.1060532208314709608618429841692590 E-27 +x^82* -2.5218920904122435678668059429207785 E-27 +x^83*  1.2457539467832514631761121316659649 E-27 +x^84* -6.1546176441900236712036669408929694 E-28 +x^85*  3.0411051132916308845009731999557020 E-28 +x^86* -1.5028717768397369996350257200319465 E-28 +x^87*  7.4279860680522329081644760189512429 E-29 +x^88* -3.6717893955394307351304683981660285 E-29 +x^89*  1.8152677927719233922578013095639624 E-29 +x^90* -8.9754810772507114316889719589853523 E-30 +x^91*  4.4384444557718439080961665330633816 E-30 +x^92* -2.1951051980324132441203842782335479 E-30 +x^93*  1.0857264522839554703474518850312616 E-30 +x^94* -5.3713051049816620023559234957667223 E-31 +x^95*  2.6569463100450525421735771247391265 E-31 +x^96* -1.3150936158893411039625390247367046 E-31 +x^97*  6.5031082874542227062257294778935637 E-32 +x^98* -3.2232822252628017181294513063989031 E-32 +x^99*  1.5906201065416840874596884648767165 E-32 +x^100* -7.8953030215835351071271084092673743 E-33 +x^101*  3.8940334244238877083713174341717584 E-33 +x^102* -1.9230987131520699822802998663703356 E-33 +x^103*  9.5400148793742290635690623957050238 E-34 +x^104* -4.7203747902964989216918644128748847 E-34 +x^105*  2.3432959042953020194125107406188183 E-34 +x^106* -1.1829537815826610344143523964787138 E-34 +x^107*  4.9057060464151121689787019274724467 E-35 +x^108* -1.1579374110263163325549928790619021 E-35 } - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread How it looks (i.θ)ₐ - by marraco - 04/18/2015, 11:20 PM RE: How it looks (i.θ)ₐ - by sheldonison - 04/19/2015, 02:40 PM RE: How it looks (i.θ)ₐ - by marraco - 04/19/2015, 08:40 PM RE: How it looks (i.θ)ₐ - by sheldonison - 04/19/2015, 11:19 PM RE: How it looks (i.θ)ₐ - by marraco - 04/20/2015, 02:35 AM RE: How it looks (i.θ)ₐ - by Gottfried - 04/20/2015, 07:53 AM RE: How it looks (i.θ)ₐ - by marraco - 04/21/2015, 02:34 AM RE: How it looks (i.θ)ₐ - by marraco - 04/21/2015, 03:20 AM RE: How it looks (i.θ)ₐ - by sheldonison - 04/21/2015, 08:23 AM RE: How it looks (i.θ)ₐ - by marraco - 04/23/2015, 04:52 PM RE: How it looks (i.θ)ₐ - by JmsNxn - 04/23/2015, 11:15 PM RE: How it looks (i.θ)ₐ - by sheldonison - 04/23/2015, 11:20 PM RE: How it looks (i.θ)ₐ - by marraco - 04/26/2015, 12:50 AM RE: How it looks (i.θ)ₐ - by sheldonison - 04/26/2015, 05:08 AM RE: How it looks (i.θ)ₐ - by marraco - 04/20/2015, 03:46 AM RE: How it looks (i.θ)ₐ - by marraco - 04/23/2015, 02:21 AM RE: How it looks (i.θ)ₐ - by marraco - 04/25/2015, 07:52 PM

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