There's a simple relationship between $J\_\nu$, Bessel functions of the first kind, and $I\_\nu$, modified Bessel functions of the first kind, namely $I\_\nu(z) = i^{-\nu} J\_\nu(iz)$. However, there doesn't seem to be any simple relationship between $Y\_\nu$, Bessel functions of the second kind, and $K\_\nu$, modified Bessel functions of the second kind. Is there an identity relating $Y\_\nu$ and $K\_\nu$ and I'm just missing it? I'm interested in arbitrary $\nu$, but a relationship that only holds for integer values would still be welcome.



Modified and unmodified Bessel functions of the second kind
$K\_{\nu}(z)$ is conventionally defined as [a linear combination of normal Bessel functions of both kinds of imaginary argument](http://dlmf.nist.gov/10.27.E8), for the simple reason that this particular linear combination of $J\_{\nu}(iz)$ and $Y\_{\nu}(iz)$ is real for real $\nu$ and positive $z$.


