So, guys, here you have to help me a bit.
I am a complete newbe to any kind of divergent summation and integral transforms.
So I was just reading a bit on Wikipedia and tried to apply the Borel summation on the half-iterate of \(e^x-1\).
I have the half iterate series \(h\) then and Wikipedia says to take
\[ \int_0^\infty e^{-t} \sum_{n=0}^\infty \frac{h_n}{n!} (tx)^n dt\]
This would make sense if the function \( e^{-t} \sum_{n=0}^N \frac{h_n}{n!} (t)^n \to 0\) for \(t\to \infty\).
But this does not happen, from a certain point the function goes rapidly to \(\infty\) and the higher I choose N the earlier this happens!
I also read something that there might be a singularity and in this case one just uses the analytic continuation along the real axis ....
But I mean numerically continue an analytic function is quite hard to do and makes no sense here.
However if I just integrate to the lowest point, I also get reasonable results:
\[h(x)=\int_0^{13.5/x} e^{-t} \sum_{n=0}^{200} \frac{h_n}{n!} (tx)^n dt\]
\(\left|h(h(x))-(e^x-1)\right|\) varies up to \(10^{-3}\) on (-0.5,0.5).
But is that how you do Borel summation???
I am a complete newbe to any kind of divergent summation and integral transforms.
So I was just reading a bit on Wikipedia and tried to apply the Borel summation on the half-iterate of \(e^x-1\).
I have the half iterate series \(h\) then and Wikipedia says to take
\[ \int_0^\infty e^{-t} \sum_{n=0}^\infty \frac{h_n}{n!} (tx)^n dt\]
This would make sense if the function \( e^{-t} \sum_{n=0}^N \frac{h_n}{n!} (t)^n \to 0\) for \(t\to \infty\).
But this does not happen, from a certain point the function goes rapidly to \(\infty\) and the higher I choose N the earlier this happens!
I also read something that there might be a singularity and in this case one just uses the analytic continuation along the real axis ....
But I mean numerically continue an analytic function is quite hard to do and makes no sense here.
However if I just integrate to the lowest point, I also get reasonable results:
\[h(x)=\int_0^{13.5/x} e^{-t} \sum_{n=0}^{200} \frac{h_n}{n!} (tx)^n dt\]
\(\left|h(h(x))-(e^x-1)\right|\) varies up to \(10^{-3}\) on (-0.5,0.5).
But is that how you do Borel summation???