From: pireddag@hotmail.com   
      
   Am 02.06.2019 um 21:25 schrieb Michael Cole:   
   > 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   
   >   
   > The only nonsingular solutions, of course, are when \lambda = l(l+1)   
   > some nonnegative integer l and \mu = m^2, \m\ \leq l and we get the   
   > generalized Legendre polys used to construct the spherical harmonics.   
   > The nonsingular solutions for various \lambda and \mu have been studied   
   > and are mathematically interesting, but my question is are there any   
   > actual physics applications?   
   >   
      
   As far as I understood the singular solutions are used in geomagnetism   
   to represent the magnetic field on portions of the Earth.   
   I know neither if this is a sensible thing to do nor whether it brings   
   good results. You should be able to find literature on the topic using   
   the search term "Spherical cap harmonics".   
      
   Giovanni   
      
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