I saw a bound of the form:
$$K_\nu(x) \leq Ce^{-x} \quad\text{for $x \geq 1$}$$
i.e. an exponential bound, somewhere, but I have no reference. Could someone tell me if this true?
2026-04-01 09:51:25.1775037085
An exponential upper bound for Bessel K function?
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We have that $\log K_\nu(x)$ is a convex function (it is a consequence of Turan's inequality for Bessel functions) and $$\lim_{x\to +\infty}\frac{\log K_\nu(x)}{x} = -1, $$ hence we have for sure: $$ \forall x\geq 1, \qquad K_{\nu}(x) \leq \left(e\cdot K_\nu(1)\right) e^{-x}.$$