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power series solutions to the legendre equation power series solutions to the legendre equation department of mathematics iit guwahati ra rks ma 102 2016 power series solutions to the legendre ...

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               Power Series Solutions to the Legendre Equation
         Power Series Solutions to the Legendre Equation
                            Department of Mathematics
                                  IIT Guwahati
                                  RA/RKS  MA-102 (2016)
              Power Series Solutions to the Legendre Equation
  The Legendre equation
      The equation
                    (1 −x2)y′′ −2xy′ +α(α+1)y = 0,                 (1)
      where α is any real constant, is called Legendre’s equation.
                   +
      When α ∈ Z , the equation has polynomial solutions called
      Legendre polynomials. In fact, these are the same polynomial
      that encountered earlier in connection with the Gram-Schmidt
      process.
      The Eqn. (1) can be rewritten as
                            2       ′ ′
                         [(x −1)y ] = α(α+1)y,
      which has the form T(y) = λy, where T(f) = (pf′)′, with
      p(x) = x2 −1 and λ = α(α+1).
                                RA/RKS  MA-102 (2016)
                Power Series Solutions to the Legendre Equation
       Note that the nonzero solutions of (1) are eigenfunctions of T
       corresponding to the eigenvalue α(α + 1).
       Since p(1) = p(−1) = 0, T is symmetric with respect to the
       inner product                   Z
                                          1
                             (f , g) =      f (x)g(x)dx.
                                         −1
       Thus, eigenfunctions belonging to distinct eigenvalues are
       orthogonal.
                                    RA/RKS   MA-102 (2016)
                  Power Series Solutions to the Legendre Equation
   Power series solution for the Legendre equation
        The Legendre equation can be put in the form
                                 y′′ + p(x)y′ + q(x)y = 0,
        where
               p(x) = − 2x             and q(x) = α(α+1), if x2 6= 1.
                            1−x2                          1−x2
        Since      1 2   =P∞ x2n for |x| < 1, both p(x) and q(x) have
                (1−x )         n=0
        power series expansions in the open interval (−1,1).
        Thus, seek a power series solution of the form
                                         ∞
                             y(x) = Xa xn, x ∈ (−1,1).
                                              n
                                        n=0
                                         RA/RKS     MA-102 (2016)
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...Power series solutions to the legendre equation department of mathematics iit guwahati ra rks ma x y xy where is any real constant called s when z has polynomial polynomials in fact these are same that encountered earlier connection with gram schmidt process eqn can be rewritten as which form t f pf p and note nonzero eigenfunctions corresponding eigenvalue since symmetric respect inner product g dx thus belonging distinct eigenvalues orthogonal solution for put q if xn both have n expansions open interval seek a xa...

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