regular tetration base sqrt(2) : an interesting(?) constant 2.76432104

[sheldonison]

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\( \hspace{48} s_0= S (x - t_0) \) where \( S(\cdot) \) denotes the Schröder function.

Hey Gottfried!

So then, the Schröder function of -infinity for base sqrt(2) from the fixed point of 2 turns out to be \( s_0 = \lim_{x\to -\infty} S (x - 2) \approx -1.31563054604637354754 \). The significance of this value is that it is also the radius of convergence of the Taylor series for the inverse Schröder function, since -infinity is the nearest singularity, so the radius of convergence would be approximately 1.3156.

Then we use the inverse Schröder function of \( -s_0 \) to get Gottfried's number.
\( z = S^{-1} (-s_0) + 2 \approx 2.7643210400001 \)
- Sheldon