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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?

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There is the identity

$$Q_\frac12 (z)=K\left(\frac{z+1}{2}\right)-2E\left(\frac{z+1}{2}\right)$$

where $K(m)$ and $E(m)$ are the complete elliptic integrals of the first and second kinds with parameter $m$. This is most easily established by using the relationships of these functions with the Gaussian hypergeometric function ${}_2 F_1\left({{a\;b}\atop{c}}\mid z\right)$.

You should then be able to use series expansions at infinity for the two elliptic integrals; however, since both elliptic integrals become complex when the parameter $m$ is greater than $1$ (in Legendre terms, if $z > 1$), I don't understand why you're expecting the coefficient $c$ to be real...

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Perhaps this is more what you look for.

For $\nu>-1$ and $x\in(1,\infty)$ we have the estimates $$Q_\nu(x)\leq\min\left(q_\nu(x-1)^{-\nu-1},x^{-\nu-1}\left(q_\nu +\frac12\log\frac{x+1}{x-1}\right)\right)$$ and $$Q_\nu(x)\geq x^{-\nu-1}\max\left(q_\nu,\frac12\log\frac{x+1}{x-1}-\gamma-\psi(\nu+1) \right)$$ where $$q_\nu=\frac{\sqrt{\pi}\Gamma(\nu+1)}{2^{\nu+1}\Gamma(\nu+3/2)}$$ and $\psi(x)=\frac{d}{dx}\log(\Gamma(x))$.

Also, in the case $\nu=1/2$ we have $q_{1/2}=\frac{\sqrt{2}\pi}{8}$.

To prove these estimates it is useful to:

(1) Use recurrence relations $$\frac{d}{dx}\left((x^2-1)Q'_\nu(x)\right)=\nu(\nu+1)Q_\nu(x)$$ $$Q'_\nu(x)=\frac{\nu+1}{x^2-1}\left(Q_{\nu+1}(x)-xQ_\nu(x)\right)=\frac{\nu}{x^2-1}\left(xQ_\nu(x)-Q_{\nu-1}(x)\right) $$

(2) Derive growth relations using (1). For example $$\frac{d}{dx}\left(x^{\nu+1}Q_\nu(x)\right)=x^{\nu}Q'_{\nu+1}(x)\tag{ <0}$$

(3) Have some convenient representations of $Q_\nu$ involving the leading term, e.g. $$Q_\nu(x)=q_\nu x^{-\nu-1}\,_2F_1\left(\frac{\nu}{2}+1,\frac{1}{2}(\nu+1),\nu+\frac32;x^{-2}\right)$$ (where $\,_2F_1$ is the hypergeometric function of Gauss).

For a details look up Proposition 3.4 in A Wiener Tauberian Theorem for Weighted Convolution Algebras of Zonal Functions on the Automorphism Group of the Unit Disc, Dahlner A. Contemp. Math no404, 2006.

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