I might be making a mistake somewhere. I know the mellin transforms don't necessarily converge; but they necessarily produce a mellin transformed function. This sort of goes "distributional theory", but not totally.
We get that--\(f(0) = 0\), and \(f'(0) = 1\), valit 1. There by we can rewrite this \(F(w) = 1/f(1/w)\). Now from here we have a half plane \(H_R\) for \(\Re(w) > R\), where \(F(w) \sim w + 1 + o(w)\), whereby \(F\) is injective on \(H_R\). Now let's do the conjugation \(G(w) = F(w+R) - R\). Now, \(G\) is holomorphic on a right half plane \(\Re(w) > 0\). What I had produced as my conclusions, were based on this.
From here, we can make another switch \(G(u) = G(1/w)\), which fully produces our expansion at infinity if \(B(G(u)) = B(u) + 1\), for a function \(B\). And \(B\) is the same thing, it is an LFT \(\ell\) applied to \(\alpha(f)\). It looks something like \(\alpha(\ell(u)) = B(u)\). Where \(\ell(G) = f(\ell)\).
Can I ask what you are trying to numerically evaluate, so I can chime in? I don't know how you're approaching it. This would be exceedingly difficult to program in. I can't think of any obvious manner of evaluating this efficiently and effectively. You have to find \(R\), you have to perform conjugations, and require taylor data at every step.
We get that--\(f(0) = 0\), and \(f'(0) = 1\), valit 1. There by we can rewrite this \(F(w) = 1/f(1/w)\). Now from here we have a half plane \(H_R\) for \(\Re(w) > R\), where \(F(w) \sim w + 1 + o(w)\), whereby \(F\) is injective on \(H_R\). Now let's do the conjugation \(G(w) = F(w+R) - R\). Now, \(G\) is holomorphic on a right half plane \(\Re(w) > 0\). What I had produced as my conclusions, were based on this.
From here, we can make another switch \(G(u) = G(1/w)\), which fully produces our expansion at infinity if \(B(G(u)) = B(u) + 1\), for a function \(B\). And \(B\) is the same thing, it is an LFT \(\ell\) applied to \(\alpha(f)\). It looks something like \(\alpha(\ell(u)) = B(u)\). Where \(\ell(G) = f(\ell)\).
Can I ask what you are trying to numerically evaluate, so I can chime in? I don't know how you're approaching it. This would be exceedingly difficult to program in. I can't think of any obvious manner of evaluating this efficiently and effectively. You have to find \(R\), you have to perform conjugations, and require taylor data at every step.