Editing Legendre function (section)

==Legendre functions of the second kind ({{math|''Q<sub>n</sub>''}})==

[[File:Mplwp_legendreQ04.svg|thumb|384px|Plot of the first five Legendre functions of the second kind.]]

The nonpolynomial solution for the special case of integer degree  <math> \lambda = n \in \N_0 </math>, and <math> \mu = 0 </math>, is often discussed separately. 
It is given by
<math display="block">Q_n(x)=\frac{n!}{1\cdot3\cdots(2n+1)}\left(x^{-(n+1)}+\frac{(n+1)(n+2)}{2(2n+3)}x^{-(n+3)}+\frac{(n+1)(n+2)(n+3)(n+4)}{2\cdot4(2n+3)(2n+5)}x^{-(n+5)}+\cdots\right)</math>

This solution is necessarily [[Singularity (mathematics)|singular]] when <math> x = \pm 1 </math>.

The Legendre functions of the second kind can also be defined recursively via [[Legendre polynomials#Definition via generating function|Bonnet's recursion formula]]
<math display="block">Q_n(x) = \begin{cases}
 \frac{1}{2} \log \frac{1+x}{1-x}  & n = 0  \\
 P_1(x) Q_0(x) - 1  & n = 1  \\
 \frac{2n-1}{n} x Q_{n-1}(x) - \frac{n-1}{n} Q_{n-2}(x)  & n \geq 2 \,.
\end{cases}</math>