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Let $J_n$ be the Bessel function of the first kind. Let $J_n^{(\max)} = \max_{x>0} J_n(x)$. What is known about the asymptotic behavior of $J_n^{(\max)}$ at large $n$? Specifically, I am looking for a lower bound. It is ok if the result only holds for integer and half-integer $n$.

(It is potentially helpful to note that $J_n^{(\max)} = J_n(j'_n)$ where $j'_n$ is the smallest positive zero of $J'_n$, i.e. the global maximum of $J_n$ occurs at the "first" local maximum.)

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The answer is given in Formula 9.5.25 of Abramowitz and Stegun, available here: https://pdfs.semanticscholar.org/1a2a/68cac86cb55abb9eb1858b3b58c4a1b16434.pdf

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