Quartic truncation of the Bessel function J0 ($J_0(\rho )\approx (1-{\rho }^2/8{)}^2$)

The regular power series has $J_0(\rho )=1-{\rho }^2/4+{\rho }^4/64-\cdots$. Its quartic truncation has a double zero at $\rho =\sqrt{8}$, giving a rough disc-well energy estimate $4{\hslash }^2/(\mu a^2)$. A truncation zero is not an exact Bessel zero or a rigorous eigenvalue bound. The double-root behavior particularly warns against treating a low-order local expansion as accurate far from its center.