  The goal of this paper is to show that evanescent plane waves are much better
at numerically approximating Helmholtz solutions than classical propagative
plane waves. By generalizing the Jacobi$\unicode{x2013}$Anger identity to
complex-valued directions, we first prove that any solution of the Helmholtz
equation on a three-dimensional ball can be written as a continuous
superposition of evanescent plane waves in a stable way. We then propose a
practical numerical recipe to select discrete approximation sets of evanescent
plane waves, which exhibits considerable improvements over standard propagative
plane wave schemes in numerical experiments. We show that all this is not
possible for propagative plane waves: they cannot stably represent general
Helmholtz solutions, and any approximation based on discrete sets of
propagative plane waves is doomed to have exponentially large coefficients and
thus to be numerically unstable. This paper is motivated by applications to
Trefftz-type Galerkin schemes and extends the recent results in [Parolin,
Huybrechs and Moiola, M2AN, 2023] from two to three space dimensions.
