= Legendre function of the second kind
{c}
{title2=$Q_n(x)$}
For nonnegative integer $n$, a second solution of the <Legendre differential equation> is obtained by <reduction of order> from $P_n$ as $P_n(x)\int dx/[(1-x^2)P_n(x)^2]$. Normalize it as $Q_0=\tfrac12\log|(1+x)/(1-x)|$ and $Q_1=xQ_0-1$, with the usual <Legendre polynomial recurrence relation> for higher degrees. Inside $(-1,1)$, $Q_n\sim-\tfrac12\log(1-x)$ near $1$ and $Q_n\sim(-1)^n\tfrac12\log(1+x)$ near $-1$. These follow from $P_n(1)=1$, $P_n(-1)=(-1)^n$ in the integral. Poles in the integral at interior polynomial zeros cancel in the product, by <reduction of order across a zero of the known solution>. On $x>1$, the integral from infinity defines the decaying normalization $Q_n(x)\sim 2^n(n!)^2x^{-n-1}/(2n+1)!$. For $n\geq1$ all bounded exterior solutions decay this way; for $n=0$ an additional constant solution remains bounded.
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