# Conjugate of Bessel functions of purely imaginary order

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.

• I would use a integral representation of $J_\alpha$: en.wikipedia.org/wiki/Bessel_function#Bessel's_integrals Mar 20, 2019 at 12:30
• Thank you, I think the second expression is the one I need to prove my conjecture. Mar 21, 2019 at 12:16