Legendre functions

Singularities of Legendre functions of the first kind (P_λ) as a consequence of symmetry

Legendre functions P_λ of non-integer degree are unbounded at the interval [-1, 1] . In applications in physics, this often provides a selection criterion. Indeed, because Legendre functions Q_λ of the second kind are always unbounded, in order to have a bounded solution of Legendre's equation at all, the degree must be integer valued: only for integer degree, Legendre functions of the first kind reduce to Legendre polynomials, which are bounded on [-1, 1]. It can be shown that the singularity of the Legendre functions P_λ for non-integer degree is a consequence of the mirror symmetry of Legendre's equation. Thus there is a symmetry under the selection rule just mentioned.