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Plane-wave expansion

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In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves: e i k ⋅ r = ∑ ℓ = 0 ∞ ( 2 ℓ + 1 ) i ℓ j ℓ ( k r ) P ℓ ( k ^ ⋅ r ^ ) , {\displaystyle 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