Plane-wave expansion

In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:

e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }({\hat {\mathbf {k} }}\cdot {\hat {\mathbf {r} }}),

where

$i$ is the imaginary unit,
$k$ is a wave vector of length $k$ ,
$r$ is a position vector of length $r$ ,
$j ℓ$ are spherical Bessel functions,
$P ℓ$ are Legendre polynomials, and
the hat $^$ denotes the unit vector.

In the special case where $k$ is aligned with the z axis,

e^{ikr\cos \theta }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta ),

where $θ$ is the spherical polar angle of $r$ .

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