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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 take the trouble to give a reference for it.

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.

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 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 take the trouble to give a reference for it.

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 take the trouble to give a reference for it.

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?

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 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 take the trouble to give a reference for it.