From: nospam@de-ster.demon.nl   
      
   Michael Cole  wrote:   
      
   > Hi all.  I am writing a monograph on orthogonal polynomials which   
   > includes discussions of physics applications.  Does anyone know of   
   > physics applications of the Legendre functions of the second kind?   
   > These are the singular (at x = 1, -1) solutions of the Legendre equation   
   > (1-x^2)y'' - 2xy' + l(l+1)y = 0$.  They can be described in terms of the   
   > Legendre polynomials by   
   >   
   >    Q_l(x) = \ln [(1+x)/(1-x) P_l(x) + (poly of order l-1)   
   >   
   > Are these just mathematical curiosities, or do they have real uses in   
   > physics? What about general solutions of the general Legendre equation   
   >   
   >     (1-x^2)y'' - 2xy + [\lambda - \mu / (1-x^2)]y = 0   
      
   What you are looking for is the toroidal-poloidal decomposition.   
   Any (modulo technicalities) divergence-free vector field,   
   so in particular a magnetic field, can be decomposed   
   into toroidal and poloidal components.   
      
   gives the formulas.   
      
   These are useful for description of magnetic fields in spherical   
   coordinates.   
   You might want to look at the reference given by wikipedia,   
   such as Chandrasekhar.   
   The singular Legendre functions of the second kind   
   can serve a basis for the toroidal and poloidal fields.   
   In particular the lowest one corresponds to the field   
   of an infinitely long conductor along the z-axis.   
      
   Best,   
      
   Jan   
      
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