Borel summation

[bo198214]

\[
**\begin{align}
\int_0^\infty e^{-x} F(\sqrt{t}x)\,dx &= \sum_{k=0}^\infty b_k t^k\\
&= t^{-1/2}\int_0^\infty e^{-ut^{-1/2}} F(u)\,du
\end{align}
\]

That's a relief, James, this formula finally works very well numerically!
This is the function under the integral t=2:
   
And at a first glance also gives very good results for the half-iterate.