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+--- Thread: Removing the branch points in the base: a uniqueness condition? (/showthread.php?tid=1075)
Removing the branch points in the base: a uniqueness condition? - fivexthethird - 03/19/2016
In many cases, when dealing with the math behind tetration, a recurring feature is the logarithm of the fixed point multiplier \( \log(\kappa) \), which I will call from here on \( \lambda \)
Since the fixed point multiplier is determined by the base, \( \lambda \) is really just the base in disguise:
\( \lambda = \log(-\text{W}(-\log(b))) \)
But all three functions have branch points that correspond to the ones in tetration's base: the inner log to 0, the productlog to \( \eta \), the outer log to 1.
Thus, I think that it's reasonable to desire the following to be the case for any reasonable tetration:
Let x > 0 and tet(x,b) be our tetration solution.
Then \( \text{tet}(x,\exp(\exp(\lambda-\exp(\lambda)))) \) analytically continues to a function without branch points in \( \lambda \)
So in other words, the branch we're on should entirely depend on what branches of those three functions we pick.