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In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: where * i is the imaginary unit, * k is a wave vector of length k, * r is a position vector of length r, * jℓ are spherical Bessel functions, * Pℓ are Legendre polynomials, and * the hat ^ denotes the unit vector. In the special case where k is aligned with the z axis, where θ is the spherical polar angle of r.
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