Iterating at fixed points of b^x

[bo198214]

Just consider the term
\( \ln(a)^n \) in \( \;^{n}b = a + \ln(a)^n \; (1-a) + \ldots \).

Öhm, a bit more explanative?

This is just a version of my definition for extending tetration from http://tetration.org/tetration_net/tetra...omplex.htm . The Taylor series of \( \;^{n}b \) taken at \( a \) has the zero term at the fixed point of course and then \( D f^n(a)= f'(a)^n = \ln(a)^n \).

This is surely true, but I dont see the connection to showing that the regular iteration at fixed point \( a \) is non-real for real arguments \( x \). Natural numbered iterations of \( b^x \) at any fixed point of course yield real values for real arguments \( x \), just \( \exp_b^{\circ n}(x) \). Which is no more true for fractional iterations.