Tommy, you know I love you. And I respect your opinion all the time. But there is a small difference in how Gottfried and I are referring to the asymptotic series. I'd just like to highlight it for you, and from what I gathered by your questions.
First of all \(g(g(z)) = e^{z}-1\) is a holomorphic function for \(\Re(z) < 0\). And equally there exists a solution \(g^*\) for \(\Re(z) > 0\). Thing is, you can't make an asymptotic series for \(g^*\), you can for \(g\). If you take \(g^{-1*}\) then you can make an asymptotic series to the right.
\(g\) is holomorphic in a half plane, and is unique; if you submit yourself to Ecalle's construction. You can modify this solution, yes; but Milnor describes Ecalle's uniqueness very well. So let's take \(g\) unique as Ecalle said.
Then Gottfried's series:
\[
|g(z)-\sum_{k=0}^N d_k z^k| < \vartheta |z|^{N+1}\\
\]
Where we can choose the constant \(\vartheta\) pointwise primarily. This series expansion is essentially what Euler did with the series expansion:
\[
h(z) = \sum_{k=1}^\infty (-1)^k k! z^k\\
\]
And the function:
\[
h(z) = \int_0^\infty \frac{e^{-pz}}{1+p}\,dp\\
\]
So.... we are trying to guess that \(d_k = O(k!)\), which Gottfried's numerical evidence is suggesting. And which, what looks like the math is suggesting. I'm just having trouble hammering a couple of the nails in, but I see in a moral sense why it's happening. I'm missing something though.
Happy to have you jump in here, Tommy
First of all \(g(g(z)) = e^{z}-1\) is a holomorphic function for \(\Re(z) < 0\). And equally there exists a solution \(g^*\) for \(\Re(z) > 0\). Thing is, you can't make an asymptotic series for \(g^*\), you can for \(g\). If you take \(g^{-1*}\) then you can make an asymptotic series to the right.
\(g\) is holomorphic in a half plane, and is unique; if you submit yourself to Ecalle's construction. You can modify this solution, yes; but Milnor describes Ecalle's uniqueness very well. So let's take \(g\) unique as Ecalle said.
Then Gottfried's series:
\[
|g(z)-\sum_{k=0}^N d_k z^k| < \vartheta |z|^{N+1}\\
\]
Where we can choose the constant \(\vartheta\) pointwise primarily. This series expansion is essentially what Euler did with the series expansion:
\[
h(z) = \sum_{k=1}^\infty (-1)^k k! z^k\\
\]
And the function:
\[
h(z) = \int_0^\infty \frac{e^{-pz}}{1+p}\,dp\\
\]
So.... we are trying to guess that \(d_k = O(k!)\), which Gottfried's numerical evidence is suggesting. And which, what looks like the math is suggesting. I'm just having trouble hammering a couple of the nails in, but I see in a moral sense why it's happening. I'm missing something though.
Happy to have you jump in here, Tommy
