12/19/2012, 01:39 PM
(This post was last modified: 12/19/2012, 10:23 PM by sheldonison.)
Welcome Lucas!
You might try posting in the math and general discussion subsection with your question too, often I just look at the Tetration and related topics section. There are formulas for the schroder coefficients for base e, which has a complex fixed point. But since the Schroder function has a complex fixed point, L~=0.318 + 1.337i, then the function is not real valued at the real axis. The Schroder function has a singularity at zero, so the regular superfunction generated from the inverse Schroder function cannot give you the sequence that you are looking for. This sequence won't work since \( \psi(0) \) is not defined, and instead the Schroder function has a singularity at zero.
\( \psi^{-1}(\psi(0))=0 \)
\( \psi^{-1}(L\psi(0))=1 \)
\( \psi^{-1}(L^2\psi(0))=e \)
\( \psi^{-1}(L^3\psi(0))=e^e \)
There is a closed form for the Schroder equation for the complex coefficients for base e, in the post you reference. There is another post here on the forum; see Mike's subpost #9.
But for real valued base e tetration, or for any base>\( e^{1/e} \), the fixed point is complex, so a Riemann mapping is required to modify the regular superfunction, \( \psi^{-1}(L^z) \) so that it becomes real valued. The Riemann mapping is complicated, and there is no simple closed form formula for real valued tetration, but there are accurate Taylor series approximations. The algorithm I use is \( \psi^{-1}(L^{z+\theta(z)}) \), where \( \theta(z) \) is a 1-cyclic function which decays to 0 as \( \Im(z) \) increases and is equivalent to the Riemann mapping. Of course, \( \theta(z) \) has a nasty singularity at integer values. There is pari-gp code to generate the Taylor series for Kneser's algorithm here.
Again, welcome, and hope this answer helps a little bit. There is a Taylor series for tetration base e below.
- Sheldon
You might try posting in the math and general discussion subsection with your question too, often I just look at the Tetration and related topics section. There are formulas for the schroder coefficients for base e, which has a complex fixed point. But since the Schroder function has a complex fixed point, L~=0.318 + 1.337i, then the function is not real valued at the real axis. The Schroder function has a singularity at zero, so the regular superfunction generated from the inverse Schroder function cannot give you the sequence that you are looking for. This sequence won't work since \( \psi(0) \) is not defined, and instead the Schroder function has a singularity at zero.
\( \psi^{-1}(\psi(0))=0 \)
\( \psi^{-1}(L\psi(0))=1 \)
\( \psi^{-1}(L^2\psi(0))=e \)
\( \psi^{-1}(L^3\psi(0))=e^e \)
There is a closed form for the Schroder equation for the complex coefficients for base e, in the post you reference. There is another post here on the forum; see Mike's subpost #9.
But for real valued base e tetration, or for any base>\( e^{1/e} \), the fixed point is complex, so a Riemann mapping is required to modify the regular superfunction, \( \psi^{-1}(L^z) \) so that it becomes real valued. The Riemann mapping is complicated, and there is no simple closed form formula for real valued tetration, but there are accurate Taylor series approximations. The algorithm I use is \( \psi^{-1}(L^{z+\theta(z)}) \), where \( \theta(z) \) is a 1-cyclic function which decays to 0 as \( \Im(z) \) increases and is equivalent to the Riemann mapping. Of course, \( \theta(z) \) has a nasty singularity at integer values. There is pari-gp code to generate the Taylor series for Kneser's algorithm here.
Again, welcome, and hope this answer helps a little bit. There is a Taylor series for tetration base e below.
- Sheldon
Code:
{tet(x)=
1.0000000000000000000
+x^ 1* 1.0917673512583209918
+x^ 2* 0.27148321290169459533
+x^ 3* 0.21245324817625628431
+x^ 4* 0.069540376139987373729
+x^ 5* 0.044291952090473304406
+x^ 6* 0.014736742096389391152
+x^ 7* 0.0086687818172252603664
+x^ 8* 0.0027964793983854596948
+x^ 9* 0.0016106312905842720722
+x^10* 0.00048992723148437733470
+x^11* 0.00028818107115404581135
+x^12* 0.000080094612538543333444
+x^13* 0.000050291141793805403695
+x^14* 0.000012183790344900091616
+x^15* 0.0000086655336673815746852
+x^16* 0.0000016877823193175389918
+x^17* 0.0000014932532485734925811
+x^18* 0.00000019876076420492745532
+x^19* 0.00000026086735600432637316
+x^20* 0.000000014709954142541901861
+x^21* 0.000000046834497327413506255
+x^22* -0.0000000015492416655467695218
+x^23* 0.0000000087415107813509359130
+x^24* -0.0000000011257873101030623176
+x^25* 0.0000000017079592672707284126
+x^26* -0.00000000037785831549229851765
+x^27* 0.00000000034957787651102163179
+x^28* -1.0537701234450015066 E-10
+x^29* 7.4590971476075052807 E-11
+x^30* -2.7175982065777348693 E-11
+x^31* 1.6460766106614471304 E-11
+x^32* -6.7418731524050529991 E-12
+x^33* 3.7253287233194685443 E-12
+x^34* -1.6390873267935902235 E-12
+x^35* 8.5836383113585680605 E-13
+x^36* -3.9437387391053843136 E-13
+x^37* 2.0025231280218870559 E-13
+x^38* -9.4419622429240650237 E-14
+x^39* 4.7120547458493713408 E-14
+x^40* -2.2562918820355970800 E-14
+x^41* 1.1154688506165369963 E-14
+x^42* -5.3907455570163504919 E-15
+x^43* 2.6521584915166818728 E-15
+x^44* -1.2889107655445536819 E-15
+x^45* 6.3266785019566604528 E-16
+x^46* -3.0854571504923359890 E-16
+x^47* 1.5131767717827405271 E-16
+x^48* -7.3965341370947514333 E-17
+x^49* 3.6269876710541876035 E-17
+x^50* -1.7757255986762984030 E-17
+x^51* 8.7098795443960546503 E-18
+x^52* -4.2692892823391563142 E-18
+x^53* 2.0950441625755281093 E-18
+x^54* -1.0278837092822587892 E-18
+x^55* 5.0468242474381763890 E-19
+x^56* -2.4780505958215521454 E-19
+x^57* 1.2173942030393317020 E-19
+x^58* -5.9816486323037815151 E-20
+x^59* 2.9402643445138969081 E-20
+x^60* -1.4455835436201850220 E-20
}