We perform a complete study
of the truncation error of the Jacobi-Anger series.
This series expands every
plane wave ${\rm e}^{i \hat{s} \cdot \vec{v}}$ in terms of
spherical harmonics
$\{ Y_{\ell, m}(\hat{s}) \}_{|m|\le \ell\le \infty} $.
We consider the truncated series where the summation is
performed over the $(\ell,m)$'s satisfying $|m| \le \ell \le L$.
We prove that if $v = |\vec{v}|$ is large enough,
the truncated series gives rise to an error lower than ϵ
as soon as L satisfies
$L+\frac{1}{2} \simeq v + C
W^{\frac{2}{3}}(K \epsilon^{-\delta} v^\gamma )\, v^{\frac{1}{3}}$
where W is the Lambert function and
$C\,, K, \, \delta, \, \gamma$ are pure positive constants.
Numerical experiments show that this
asymptotic is optimal. Those results are
useful to provide sharp estimates for the
error in the fast multipole method for
scattering computation.