eta as branchpoint of tetrational

[sheldonison]

But I am really amazed how quickly you develop your code for new functions!

Ok, here are the theta(z) values, for the Fatou(z) function,
   
I used these two equations, which are equal.
\( \text{Fatou_{\sqrt{2}}(z)=\text{Usexp_{\sqrt{2}}(z+\theta_U(z)+k) \)
\( \text{Fatou_{\sqrt{2}}(z)=\text{Lsexp_{\sqrt{2}}(z+\theta_L(z)) \)
Next, calculate thetau via this equation, near imag(z)=2i, by doing a Fourier series over a unit length, starting with thetaL(z)=0.
\( \theta_U(z)+k=\text{Usexp}^{-1}(\text{Lsexp(z+\theta_L(z))-z \)

Now, using the theta_u from above, near imag(z)=15i, by doing a Fourier series over a unit length.
\( \theta_L(z)=\text{Lsexp}^{-1}(\text{Usexp(z+k+\theta_U(z))-z \)
Repeat one more time, and the two series are consistent, and accurate to more than 67 decimal digits accuracy. Be careful to remember that thetaU decays to zero as imag(z) increases, and thetaL decays to zero as imag(z) decreases. \( \theta_U(z) = \sum_{n=1}^{\infty} a_n\exp(2n\pi i z) \) and \( \theta_L(z) = \sum_{n=1}^{\infty} a_n\exp(-2n\pi i z) \). I verified that there is a very large overlap where \( \text{Usexp(z+\theta_U(z)+k)=\text{Lsexp}(z+\theta_L(z)) \), accurate to 67 decimal digits, which is the accuracy of the superfunctions that I used.

Theta(z) results are posted at 8.5715741i, half the average of the two Periods, where the two theta(z) values have equal magnitudes, and both are equally small. Note the a1 terms have nearly identical magnitudes, with a1_U ~= -conj(a1_L), to 50 digits accuracy. This is where imag(f(z)) goes from 4- to 2+ as z goes from -inf to +inf. At the real axis, thetaU has a singularity, at integer values of z. At the real axis, thetaL has very small terms, with a1~=-2E-50. ThetaL has a singularity, that I don't understand as well, around imag(z) = average of the two Periods.

This method would also probably work for complex bases, where one fixed point is repelling, and the other fixed point is attracting. This might provide a mechanism to verify Mike's results for complex bases, posted here.. Also, this post is closely related to my attracting fixed point thread, where I also posted theta(z) mappings for B=sqrt(2).

Code:

ThetaU(z), starting with the constant "k"; the superfunctions used in these calculations were 67 digits accurate
k=5.284046911275929509562319765 + 1.046500431344003802826235228*I
a1= 5.132787355776188711993056404 E-27 - 3.625724477536479451525596757 E-25*I
a2= 1.511792461497588143794162538 E-50 - 6.197254624020296328476658294 E-49*I
a3= 4.741105856335653880260054426 E-74 - 1.519169304975871323751503497 E-72*I
a4= 1.570350059077682233462666141 E-97 - 4.321977227564835332037557804 E-96*I
a5= 5.396718213010231485424958665 E-121 - 1.334210621916187903869149050 E-119*I
For ThetaL, I drove the "k" constant to zero.
KL=~0=-1.3080789967888603754 E-68 + 2.880161851 E-67*I
a1= -5.132787355776188711993056404 E-27 - 3.625724477536479451525596757 E-25*I
a2=  8.268182523502499158985049574 E-51 + 2.060888780809025676827314260 E-49*I
a3= -3.994533474813316741919755528 E-75 - 1.060362818161862613179451797 E-73*I
a4= -2.431964709922672038589803344 E-99 + 4.894555977076092256343445907 E-98*I
a5=  5.436928486837832888855701860 E-123 - 2.185866273307809932369827910 E-122*I