[Math] complex conjugate of Bessel function

Question

This is probably a simple question, however, how does one take a complex conjugate of the Bessel function,$$
J_1(z),\quad
z\in \mathbb{C}$$
I am asking because I am interested in calculating
$$
|J_1(z)|^2=J_1(z)\cdot {\bar{J_1}}(z)
$$
where the bar above $J_1$ denotes complex conjugation.
In general for a complex number we can write
$$
z=x+iy,\quad \bar{z}=x-iy
$$
If you're wondering why this is relevant, well for example, an integral of the form
$$
\int z J^2_1(z)dz=\frac{1}{2}z^2 (J^2_1(z)-J_0(z)J_2(z))
$$
only when $z\in \mathbb{R}$
however , I am interested in calculating
$$
\int z |J_1(z)|^2 dz
$$
when $z\in \mathbb{C}$,
where $|J_1(z)|^2=\bar{J}_1\cdot J_1$. (if anybody knows how to do this integral, feel free to let me know as well).

Thanks!

Answer 1

If a holomorphic function $f$ maps reals to reals, then it satisfies the equality $\overline{f(z)} = f(\overline z)$.