  The dynamic hyperpolarizability of a particle bound by the one-dimensional
$\delta$-function potential is obtained in closed form. On the first step, we
analyze the singular structure of the non-linear response function as given by
the sum-over-state expression. We express its poles and residues in terms of
the wave-number $k$. On the second step, we calculated the frequency dependence
of the response function by integration over $k$. Our method provides a unique
opportunity to check the convergence of numerical methods, and is in a perfect
agreement with the static and high frequency limits obtained by different
theories. The former is obtained using the approach of Swenson and Danforth (J.
Chem. Phys. {\bf 57}, 1734 (1972)). The asymptotic decay is studied using the
method of Scandolo and Bassani (Phys. Rev. B {\bf 51}, 6925 (1995)). Its
extension to the case of quadrupole polarizability reveals a universal (not
dependent on the choice of the system) asymptotic behavior of the
hyperpolarizability.
