Estimates for Bessel functions

Question

Are there known (above and/or below) estimates for the Bessel function $K_a(x)$? I am interested in the case where $x$ is positive real and $a$ is complex.

Answer 1

For $x > 0$, $$ K_a(x) = \int_0^\infty \exp(-x \cosh(t)) \cosh(a t)\; dt $$ If $a = b + i c$ where $b$ and $c$ are real, $$ \cosh(a t) = \cosh(b t) \cos(c t) + i \sinh(b t) \sin(c t) $$ Estimate real or imaginary part, do the integration and you get a corresponding estimate for $K_a(x)$. For example, since $-1 \le \cos(c t) \le 1$, $$ |\text{Re}(K_a(x))| \le \int_0^\infty \exp(-x \cosh(t)) \cosh(b t)\; dt = K_b(x)$$ while since $-1 \le \sin(c t) \le 1$ and $|\sinh(b t)|\le \cosh(b t)$, you get the same estimate for $|\text{Im}(K_a(x))|$.

I'm sure there are better estimates.