Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

Solid harmonics

Solid harmonics


In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the regular solid harmonics R^m_ell(mathbf{r}), which vanish at the origin and the irregular solid harmonics I^m_{ell}(mathbf{r}), which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:

 
R^m_{ell}(mathbf{r}) equiv sqrt{frac{4pi}{2ell+1}}; r^ell Y^m_{ell}(	heta,varphi)
 
I^m_{ell}(mathbf{r}) equiv sqrt{frac{4pi}{2ell+1}} ; frac{ Y^m_{ell}(	heta,varphi)}{r^{ell+1}}

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