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In NIST the following bound is claimed: $|J_\nu(x)|\le 1$ for all $x,\nu\ge0$. This is trivial for integer $\nu$, and it is pretty easy to prove a bound with 1 replaced, say, by 2. Does anyone have a reference for the sharp bound 1?

EDIT: let me clarify that the bound is true, and it follows e.g. by applying a bound by Landau $|J_\nu(x)|\le 0.79|x|^{-1/3}$ for $x>2$, plus a more standard estimate for $x<2$. I am looking for a simpler and more natural proof; I guess it should exist, since NIST does not even bother to give a reference for it.

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  • $\begingroup$ do u know what happens in this paper (I don't)? $\endgroup$
    – mathworker21
    Commented Mar 30 at 13:15
  • $\begingroup$ Yes, it's a nice paper on sharp asymptotic bounds. Let me add that the bound can be proved by applying results from a related paper by Landau. But the claim from NIST is earlier and should be easier to prove (I guess!) $\endgroup$
    – Piero D'Ancona
    Commented Mar 30 at 13:17
  • $\begingroup$ Indeed a quadratic exponential decay of $J_\nu$ seems a bit too much. Certainly the estimate refers to a restricted interval of $x$ $\endgroup$
    – Piero D'Ancona
    Commented Mar 30 at 17:59
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    $\begingroup$ Watson p. 406 equation (10) and preceding statement. Watson cites Lommel, Münchener Abh. xv. (1886), pp. 561-563, metadata. There is also a related note on p. 152. $\endgroup$
    – Georg Essl
    Commented Mar 31 at 13:06
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    $\begingroup$ This is it, thank you $\endgroup$
    – Piero D'Ancona
    Commented Apr 1 at 6:17

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