One simplification for Bessel Function (K)

Question

My question is if it is possible simplify the following expression involving Bessel function $K_v(z)$:

$$K_v(z)+K_{\overline{v}}(z)$$

where $v\in \mathbb{C}$ and $\overline{v}$ is the conjugate of $z$.

Specifically, I need when $v=1/2+i\,y$.

Answer 1

You might find this useful: $K_{\overline{v}}(z) = \overline{K_v(\overline{z})}$. In particular, if $z$ is real, your expression is twice the real part of $K_v(z)$.