x↑↑x = -1

[sheldonison]

For a complex base, tet is not a real valued function at the real axis. But if the base is real, then yes, the two theta functions agree naturally due to Schwarz reflection involving L and L*. So the complex tet function naturally becomes the Kneser solution for a real valued base.

Im aware of that.

I mean mapping a continu line on the complex plane to the positive reals.

For instance mapping f(4+3i) to g(1) as a point example.

... it seems weird that 2 analytic functions can agree on the real line , yet be different off the real line...

so f(1) = g(1) = a , f(2) = g(2) = b , ... for all positive reals ... yet f and g are different ??

regards

tommy1729

They don't disagree, they just become less accurate. In particular, theta1 has a singularity at the real axis, so you can't use it in the bottom half of the complex plane, and it isn't very accurate right at the real axis, but you can use theta2 in the bottom half of the complex plane, but not too close to the real axis, where it is not that accurate, and not the upper half of the complex plane. But the two theta mappings agree with the third function, which is a unit radius Taylor series, in their respective halves of the complex plane. The tet 3+I Taylor series in post#7, is accurate to a little better than 13 decimal digits, inside a circle radius=0.85, and agrees with the theta mappings to better than 13 decimal digits if abs(imag(z))>0.2i. So all three functions are in excellent mutual agreement wherever they converge well, and one can easily double the pari-gp precision, or quadruple it, so results are accurate and consistent to 32 decimal digits, or 66 decimal digits.

The motivation for complex tetration is that it is conjectured that all of the following become analytic functions, with complex tetration, where z is the tetration base. For example, the half iterate of 0 becomes an analytic function with respect to the tetration base.
\( f(z)=\text{tet}_z(-0.5)\;\;f(z)=\text{tet}_z(i)\;\;f(z)=\text{tet}_z(-i+1)\;\; \) or any other value, as you vary the tetration base z

Numeric evidence also supports this conjecture; see post#9. And in fact, there are bases on the ShellThron boundary that don't have the required analytic upper Abel function and theta mapping, but we can accurately calculate tetration for those bases indirectly by assuming tetration is analytic as you vary the base.

Sometimes a couple of pictures can help The first picture is tet base=3+i at the real axis from -1.99 to +2. The second picture is a complex plane graph, from -4 to +16 in the reals, and -6 to +4 in the imaginary, with grids every 2 units. In the lower graph, you can see how the upper and lower halves of the complex plane have different pseudo periods. Yet the conjecture is the function is analytic in the upper and lower halves of the complex plane, with singularities at the -integers, less than or equal to -2. You can see the zero at -1, and the logarithmic singularity at -2, -3. The cutpoints for z<-2 are kind of random ... I find the complex plane graphs of some of these tetcomplex functions to be truly beautiful.