Let's take "analytic" to mean "determined by its maclaurin series". It's not surprising that polynomials are analytic. Look at the definition of a polynomial. I

Let's take "analytic" to mean "determined by its maclaurin series". It's not surprising that polynomials are analytic. Look at the definition of a polynomial. It's already in "power series" form, in fact there are only finitely many terms! As for the other examples, here is one explanation for why they are ubiquitous in physics. Essentially, the class of analytic functions is closed under "solving differential equations". So that's why sin/cos/exp/bessel are analytic - they are solutions to differential equations with constant/polynomial coefficients (we already know that constants and polynomials are analytic). That's why many functions found in physics are analytic - they are created from other analytic functions via a differential equation. https://math.stackexchange.com/a/190167 https://math.stackexchange.com/a/190167 P.S: Regarding your last statement "almost all smooth functions are almost polynomial", something much stronger is true: Every continuous function is almost a polynomial. https://en.wikipedia.org/wiki/Stone–Weierstrass_theorem https://en.wikipedia.org/wiki/Stone–Weierstrass_theorem