TPID 4

[sheldonison]

Then how do you explain the violation of the chain law ?

regards

tommy1729

The flaw in your proof is that k is an integer, but you state k as a real number. "sexp ' (w+k) = exp^[k] ' (sexp(w)) * sexp ' (w) = 0". For k as a fraction, the chain rule does not apply. As I stated, the initial approximation used for tet_alt(z) as a seed before iterating the theta(z) mapping from the secondary fixed point is:

\( \text{tet}_{\text{alt}}(z) \approx f(z) = \text{tet}_{\text{primary}}(z - \frac{sin(2\pi z)}{2\pi}) \)

f(z) is not tet_alt(z), but merely an approximation using an analytic entire theta mapping from the primary fixed point tetration solution. But most important it shows the flaw in your theorem. This approximation also has f'(n) and f''(n)=0 at all integers>=-2, just like tet_alt, and f(z+1)=exp(f(z)), just like tet_alt. And it is analytic at the real axis. So f(z) is also an example of a function that shows the chain law can only be used for integers.

Unlike the tet_alt(z) function, f(z) has an infinite number of singularities in the upper and lower halves of the complex plane, where \( z+\theta(z) \) is a negative integer<=-2. Tet_alt is generated from the equivalent of a Riemann mapping from the secondary fixed point, which guarantees it is analytic in the upper and lower halves of the complex plane, see the alternate fixed point post, #19. for details. But the important thing for our discussion of the generalized tpid#4 is the \( \theta(z) \) below is an example showing that \( \theta(z) \) need not be entire....

...how do we prove that \( \theta(z) \) has to be entire? Is there the possibility that theta(z) has singularities, but sexp(z+theta(z)) is still analytic?

\( \text{tet}_{\text{alt}}(z) = \text{tet}_{\text{primary}}(z +\theta(z)) \)
As Mike pointed out, this \( \theta(z) \) is the example that shows \( \theta(z) \) need not be entire, yet tet(z+theta(z)) is still analytic in the upper/lower halves of the complex plane.