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.)



Asymptotic behavior of maximum of bessel function
The answer is given in Formula 9.5.25 of Abramowitz and Stegun, available here:
<https://pdfs.semanticscholar.org/1a2a/68cac86cb55abb9eb1858b3b58c4a1b16434.pdf>


