Hyperbolic Debye asymptotic for Legendre polynomials

For fixed

ξ > 0

,

P_{n} (cosh ξ) \sim \frac{e ^{(n + 1/2) ξ}}{2 πn s i n h ξ} .

Table of contents
- Legendre boundary layer at x equals one Hyperbolic Debye asymptotic for Legendre polynomials

Legendre boundary layer at x equals one ( $P_{n} (1 + ν / n^{2}) \to I_{0} (2 ν)$ )

The endpoint Laplace method for a Legendre polynomial ceases to localize when

n x - 1 = O (1)

. On the scale

x = 1 + ν / n^{2}

, its real integral representation gives

P_{n} (1 + ν / n^{2}) ⟶ \frac{1}{π} \int_{0}^{π} e^{2 ν c o s θ} d θ = I_{0} (2 ν) .

The Modified Bessel function of the first kind equals one at

ν = 0

and has large-

ν

behavior

e^{2 ν} / 2 π 2 ν

, 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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