Editing Legendre function (section)

==Singularities of Legendre functions of the first kind ({{math|''P''<sub>''λ''</sub>}}) as a consequence of symmetry ==
Legendre functions {{math|''P''<sub>''λ''</sub>}} 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 {{math|''Q''<sub>''λ''</sub>}} 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<ref>{{Cite journal |last=van der Toorn |first=Ramses |date=4 April 2022 |title=The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre's Equation |journal=Symmetry |language=en |volume=14 |issue=4 |pages=741 |doi=10.3390/sym14040741 |bibcode=2022Symm...14..741V |issn=2073-8994 |doi-access=free }}</ref> that the singularity of the Legendre functions  {{math|''P''<sub>''λ''</sub>}}  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.