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Let $I_1(z)$ be the Bessel function of the first order with purely imaginary argument.

Can we explicitly bound $I_1$ on $[0,x]$, where $x>0$ is a real number in terms of $x$?

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Note that $$I_1(iz)=i\,J_1(z)$$. Then see these. –  J. M. Dec 5 '11 at 9:54
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