Removing the branch points in the base: a uniqueness condition? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- 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.