Weyl expansion
In physics, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. In a Cartesian coordinate system, it can be denoted as e − j k 0 r r = 1 j 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ d k x d k y e − j ( k x x + k y y ) e − j k z | z | k z {\displaystyle {\frac {e^{-jk_{0}r}}{r}}={\frac {1}{j2\pi }}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dk_{x}dk_{y}e^{-j(k_{x}x+k_{y}y)}{\frac {e^{-jk_{z}|z|}}{k_{z}}}} , where k x {\displaystyle k_{x}} , k y {\displaystyle k_{y}} and k z {\displaystyle k_{z}} are the wavenumbers in their respective coordinate axes: k 0 = k x 2 + k y 2 + k z 2 {\displaystyle k_{0}={\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}}} .