= Legendre boundary layer at x equals one
{c}
{title2=$P_n(1+\nu/n^2)\to I_0(\sqrt{2\nu})$}
The endpoint <Laplace method> for a <Legendre polynomial> ceases to localize when $n\sqrt{x-1}=O(1)$. On the scale $x=1+\nu/n^2$, its real integral representation gives
$$
P_n(1+\nu/n^2)\longrightarrow \frac1\pi\int_0^\pi e^{\sqrt{2\nu}\cos\theta}\,d\theta=I_0(\sqrt{2\nu}).
$$
The <Modified Bessel function of the first kind> equals one at $\nu=0$ and has large-$\nu$ behavior $e^{\sqrt{2\nu}}/\sqrt{2\pi\sqrt{2\nu}}$, matching the <Hyperbolic Debye asymptotic for Legendre polynomials>. This <boundary layer> resolves an increasingly steep function at an endpoint where its value remains fixed.
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