03/19/2016, 10:44 AM
(This post was last modified: 03/19/2016, 11:04 AM by fivexthethird.)
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.
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.
