# Is a Macdonald function a Bessel function with imaginary argument??

I mean that

$$K_{a} (x)= CJ_{a}(ix).$$

Here $C$ is a complex number, and $a$ is real.

So is the Macdonald function a Bessel function in disguise (or proportional) of complex argument??

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In fact $$K_a(x) = \frac\pi2 i^{a+1} H^{(1)}_a(ix) = \frac\pi2 i^{a+1} [J_a(ix) + i Y_n(i x)]$$ so it is closer related to the Hankel $H^{(1)}$ than to the Bessel function (of course the Hankel function is just a linear combination of the two Bessel functions $J$ and $Y$).