Modified Bessel function with imaginary index is purely real?

[perishingtardi]

I'm trying to decide if the modified Bessel function $$K_{i \beta}(x)$$ is purely real when $\beta$ and $x$ are purely real. I think that is ought to be. My reasoning is the following:

$$\left (K_{i \beta}(x)\right)^* = K_{-i \beta}(x) = \frac{\pi}{2} \frac{I_{i \beta}(x) - I_{-i \beta}(x)}{\sin(-i \beta\pi)} = \frac{\pi}{2} \frac{I_{i \beta}(x) - I_{-i \beta}(x)}{-\sin(i \beta\pi)} = \frac{\pi}{2} \frac{I_{-i \beta}(x) - I_{i \beta}(x)}{\sin(i \beta\pi)} = K_{i \beta}(x).$$

I have used here the fact that sine is an odd function and the definition of the K function in terms of the I function. So it seems that the complex conjugate of K is K itself in this case.However, Mathematica is telling me that K is imaginary if $x<0$. Have I made a mistake somewhere? Thanks

 

[perishingtardi]

I think I know what the problem is. The first equality is wrong, i.e., $\left( K_{i \beta}(x) \right)^*$ is not simply $K_{-i\beta}(x)$.