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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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