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