Hello all I am having a bit of confusion in regard to the following;
I understand that when we are working with ODE of the form
$$p_o(x)y''(x)+p_1(x)y'(x)+p_2(x)y(x)=0$$ and we are considering some singular point say $x_o$.
I know then for if all p polynomials, we can use the limit method to determine if it is regular or irregular singular point.
But also I know it must hold in general for all analytic functions and the limit method can only be used with polynomials.
So my question is, how can I know if something is analytic. I know the definition of it, i.e. , if it has a convergent power series at $x_o$, but surely we are not supposed to construct a taylor series for each function we see to do this?
Any advice? Thank you