Originally Posted by

**Normandy6644**
I was thinking about sine and cosine too. I don't see anything wrong with them being infinitely differentiable. I don't know enough about Bessel functions to say anything about them though. I bet there are probably some conditions an infinitely differentiable function would satisfy, but I don't know what they are.

This isn't perfect (see below), but to prove a function is infinitely (continuously) differentiable, say in some open interval, it is sufficient to prove that it has a convergent power series in that interval (the power series is then the Taylor series, and you can read off the derivatives). In particular, any polynomial works, as does the exponential, and sine and cosine.