As we are talking about functions, we know that there are functions that are infinitely differentiable. An analytic function is a function that its Taylor series at any point x_0 in its domain converges to the function itself for x of x_0. In other words, if a function is analytic at c, then the function can be expanded to a  power series around c  has a positive radius of convergence. For example, if f(x) = e^x is analytic at x=0, then we are able to write out the power series representation ∑((x^n)/n!) as n≥0 near 0. So, the major difference between analytic functions and infinitely differentiable functions is that we can always write out a power series representation for analytic functions but not every infinitely differentiable function has a power series representation on every x in its domain.