How common/rare are analytic functions in applied math?

Question

How rare or how common are the analytic (differentiable infinitly many times) functions in real applied math? Is it like a "this almost never happens" or is it "most of the time you can approximate a physics experiment with analitic functions (and if you failed, you are bad at math)"? I ask because there are so many nice properties of analytic functions in complex analysis. But are they relevant? Seems like "differentiable forever" is too strong of a limitation.

Answer 1

First, as in comments, "analytic" is universally (in this year) a definitely stronger condition than "indefinitely/infinitely differentiable". Specifically, all smooth functions have finite-order Taylor-Maclaurin series with remainder, but (in a Baire category sense) most are not equal to their infinite power-series expansion. Having a correct infinite power series expansion is the definition of "analytic".

On another hand, "perturbation methods" (often exactly just heuristics) in many pure, applied, physics, and other situations, might seem to depend on an assumption of analyticity, but really only depend (in all applications I've seen) on finite Taylor-Maclaurin expansions... admittedly with values in spaces of functions, so there are still some non-trivial issues.

In many formulaic scenarios, this "mere differentiability" in a parameter is at least very plausible, and often immediately provable. But, now-and-then, the more speculative part involves an implicit assumption that the error term at a specific degree of expansion is controllable, as a function-valued thing, not just as a number. This is what is often not justified, and often because it is very difficult to do so. Still, such a heuristic does provide information. Further, although such estimates without genuine error terms cannot be entirely trusted, sometimes the true estimate does make things work, but we can't prove that estimate, so we can't prove that things work... though they do. :)