From: patzermike.mc@gmail.com   
      
   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?   
      
   Most treatments of this stuff in the literature either omit complete   
   proofs (it can be shown that ... ) or else use unnecessarily advanced   
   methods. I am trying to write a nice treatise that combines completeness   
   of proof and explanation with simplicity of method and accessibility to   
   a reader who hasn't studied much theory of DEs or functional analysis.   
   I also intend to discuss general Gegenbaur and Jacobi polynomials (with   
   simpler and more elementary proofs than are usually found in the   
   literature).  Does anyone know nice physics applications of these   
   polynomials that are simple enough to present in a short monograph?   I   
   will conclude with a brief discussion of orthogonal polynomials in more   
   than one variable.  The polynomials in two variables orthogonal on the   
   unit disk in R^2 have found applications in tomography and lense optics.   
    General orthogonal polynomials in many variables are mathematically   
   quite interesting, but haven't been studied much since they don't seem   
   to have apparent applications.   
      
        Thanks for all suggestions.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)