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 $\overline{f(z)} = f(\overline z)$.

Answer 2

By the Paley-Wiener Theorem, the Bessel functions $J_n(t)$ are entire functions mapping $\mathbb{R}$ to $\mathbb{R}$, since they are the inverse Fourier transforms of compact-supported, even real functions. By the Schwarz reflection principle it follows that: $$ \overline{J_n(z)} = J_n(\overline{z}).$$