I would like to find a relation between $J_{i\nu}(x)$ and $J^*_{i\nu}(x)$ where $J$ are Bessel functions of the first kind, $*$ denotes the conjugate, and $\nu,x\in \mathbb{R}$ so that the functions have a purely imaginary order.

My intuition is that $J^*_{i\nu}(z)=J_{-i\nu}(z^*)$ if $z\in\mathbb{C}$ so in my case I will have $J^*_{i\nu}(x)=J_{-i\nu}(x)$ but I haven't found any proof for this result.

Any help is appreciated.



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