I am given the Bessel function of first kind of order $n$, $$J_n(\beta)= \frac{1}{\pi} \int_{0}^\pi \cos(\beta \sin x-nx)dx.$$ $n$ is positive integer. It's value is known as $J_n(\beta)\geq0.01$ for a particular $\beta$, meaning I need to find out the maximum $n$ for which the function doesn't fall below $0.01$. How can I determine the order $n$ from this?
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If $n$ is real, $J_n$ has infinitely many positive real zeros. Thus, your inequality cannot hold with any particular real $n$ for all $\beta \in \mathbb R$. – Gary Aug 11 '21 at 07:04
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Here $n$ is positive integer only and $\beta$ is also given – Elin Das Aug 12 '21 at 08:02
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Are you aware of the fact that a positive integer is also a real number? Thus, $J_1,J_2,J_3,\ldots$ all have infinitely many positive real zeros. – Gary Aug 12 '21 at 08:03
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I want to find out $n$ for a given $\beta$. – Elin Das Aug 12 '21 at 08:06
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I do not think there is a simple way to solve this problem. The maximal $n$ will depend in a nontrivial way on $\beta$. – Gary Aug 12 '21 at 14:31