Does anyone know of any examples of functions that can be analytically continued (or continued using another method), and have a radius of convergence of zero? I have read before that something like conformal mapping can be used to continue functions with a 0 radius of convergence, but I have been unable to find any specific examples. I would appreciate it if anyone could share any examples or links to examples of functions that fit the criteria.
Update: I've found a few examples to get started with. Somos gave a link that shows that
$$\sum_{n=0}^\infty (-1)^nn!x^n$$
can be continued to
$$e^{\frac{1}{x}}\int_{0}^{x}\frac{e^{-\frac{1}{t}}}{t}dt$$
as shown in the graph below
The other example I found is $\sum_{k=0}^{\infty}\left(\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\left(2l+2\right)^{k}}{\left(2l+1\right)!}\right)x^{k}$ can be continued to
$$\int_{0}^\infty e^{-t} e^{st}\sin(e^{st})dt$$