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It is known that, for $z \gg \nu^2$, we have the asymptotic bound

$$\displaystyle |J_{\nu}(z)| \leqslant C_{\nu}|z|^{-1/2},$$

where $J_{\nu}$ denotes the Bessel function of the first kind, and $C_{\nu}$ denotes some positive constant depending on $\nu$. There are also different asymptotics for $z \ll 1$. I'd like to know if, for sufficiently large $z$, we can do better. For instance, suppose that $z \gg 10^{100}$. Can we do any better than the above bound, or is the above bound optimal? Can we take $C = 5000$ and the power of $|z|$ to be $-101/200$, for example?

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    Where did you see an exponent $-3/2$ ? – 2017-02-21
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    Sorry, I meant $-1/2$ -- fixed. – 2017-02-21

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No, the Bessel functions are oscillatory, but have an envelope decreasing in $1/\sqrt z$. This is very well confirmed by the half-integer case, which is exactly the ratio of a sinusoid over the square root.

The bound $\sqrt{\dfrac2{\pi z}}$ is tight.