Editing Legendre function (section)

== Legendre's differential equation ==
The '''general Legendre equation''' reads
<math display="block">\left(1 - x^2\right) y'' - 2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right] y = 0,</math>
where the numbers {{math|''λ''}} and {{math|''μ''}} may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when {{math|''λ''}} is an integer (denoted {{math|''n''}}), and {{math|1=''μ'' = 0}} are the Legendre polynomials {{math|''P<sub>n</sub>''}}; and when 
{{math|''λ''}} is an integer (denoted {{math|''n''}}), and {{math|1=''μ'' = ''m''}} is also an integer with {{math|{{abs|''m''}} < ''n''}} are the associated Legendre polynomials. All other cases of {{math|''λ''}} and {{math|''μ''}} can be discussed as one, and the solutions are written {{math|''P''{{su|p=''μ''|b=''λ''}}}}, {{math|''Q''{{su|p=''μ''|b=''λ''}}}}. If {{math|1=''μ'' = 0}}, the superscript is omitted, and one writes just {{math|''P<sub>λ</sub>''}}, {{math|''Q<sub>λ</sub>''}}. However, the solution {{math|''Q<sub>λ</sub>''}} when {{math|''λ''}} is an integer is often discussed separately as Legendre's function of the second kind, and denoted {{math|''Q<sub>n</sub>''}}.

This is a second order linear equation with three regular singular points (at {{math|1}}, {{math|−1}}, and {{math|∞}}). Like all such equations, it can be converted into a [[hypergeometric differential equation]] by a change of variable, and its solutions can be expressed using [[hypergeometric function]]s.