Hi I am trying to integrate:
$$ \int_0^R x|J_1(bx)|^2 \, dx,\quad R>0, b\in \mathbb{C} $$ where $|J_1|^2=\bar{J}_1 J_1$ where $\bar{J_1}$ means complex conjugation of the bessel function.
My attempt was $$ \int_0^R x J_1(bx) \bar{J_1}(bx) \, dx=\int_0^R x J_1(bx) J_1(\bar b x) \, dx. $$ if we now define $c\equiv \bar b$ we can write $$ \int_0^R x J_1(bx) {J_1}(\bar bx) \, dx=\int_0^R x J_1(bx) J_1(c x) \, dx, $$ but this integral can be done and is given by: $$ \int_0^R x J_1(bx) J_1(c x)dx=R\frac{cJ_1(bR)J_0(cR)-bJ_0(bR)J_1(cR)}{b^2-c^2}. $$ Is this correct, if so, is there anyway to simplify this? If not, how can we do this integral? Is it possible? I feel like if it is not correct, maybe it is not okay to use $c\equiv \bar b$ and proceed with the integration. Thanks!!