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Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$.

One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental functions, the Bateman Manuscripta Project Volume 1 .

I'm looking for a more precise statement. Namely, I would like to know if one can prove an upper bound for $Q_{1/2}(u)$ of the form $c u^{-3/2}$, where $c$ is an explicit real number.

Where can I find this, or how can I derive this?

2 Answers 2