Solid harmonics

In <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the solid harmonics are solutions of the <a href="/facts/Laplace_equation/8n1YkmFS">Laplace equation</a> in <a href="/facts/Spherical_polar_coordinates/5cVnNof3">spherical polar coordinates</a>, assumed to be (smooth) functions 
 
 
 
 
 
 R
 
 
 3
 
 
 →
 
 C
 
 
 
 {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }
 
. There are two kinds: the regular solid harmonics 
 
 
 
 
 R
 
 ℓ
 
 
 m
 
 
 (
 
 r
 
 )
 
 
 {\displaystyle R_{\ell }^{m}(\mathbf {r} )}
 
, which are well-defined at the origin and the irregular solid harmonics 
 
 
 
 
 I
 
 ℓ
 
 
 m
 
 
 (
 
 r
 
 )
 
 
 {\displaystyle I_{\ell }^{m}(\mathbf {r} )}
 
, which are singular at the origin. Both sets of functions play an important role in <a href="/facts/Potential_theory/3sXGIcDA">potential theory</a>, and are obtained by rescaling <a href="/facts/Spherical_harmonics/bn98Gv2L">spherical harmonics</a> appropriately:

R
          
            ℓ
          
          
            m
          
        
        (
        
          r
        
        )
        ≡
        
          
            
              
                4
                π
              
              
                2
                ℓ
                +
                1
              
            
          
        
        
        
          r
          
            ℓ
          
        
        
          Y
          
            ℓ
          
          
            m
          
        
        (
        θ
        ,
        φ
        )
      
    
    {\displaystyle R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi )}

I
          
            ℓ
          
          
            m
          
        
        (
        
          r
        
        )
        ≡
        
          
            
              
                4
                π
              
              
                2
                ℓ
                +
                1
              
            
          
        
        
        
          
            
              
                Y
                
                  ℓ
                
                
                  m
                
              
              (
              θ
              ,
              φ
              )
            
            
              r
              
                ℓ
                +
                1
              
            
          
        
      
    
    {\displaystyle I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1}}}}

Solid harmonics open-in-new

Solid harmonics