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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.

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up vote 3 down vote accepted

$K_{\nu}(z)$ is conventionally defined as a linear combination of normal Bessel functions of both kinds of imaginary argument, for the simple reason that this particular linear combination of $J_{\nu}(iz)$ and $Y_{\nu}(iz)$ is real for real $\nu$ and positive $z$.

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There is, but I think you need to include an extra $J_\nu$ or $I_\nu$. See the formulas here: http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/27/01/ (in particular #3 and #4 from the top).

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