The orthogonality of the Bessel function of first kind order zero states that:
$$ \int_0^1 rJ_0(R_nr)J_0(R_mr)dr={J_1^2(R_n)\over 2} \tag 1 $$
where $R_n$ is the $n$-th root of the Bessel function and $R_m=R_n$. I am trying to use this property to solve the following integral:
$$ \int_0^{r_{max}} {r \over r_{max}} J_0\Bigg({R_nr \over r_{max}}\Bigg)J_0\Bigg({R_nr \over r_{max}}\Bigg)dr=? \tag 2 $$
Seems to me that if I use the substitution:
$$ \rho = {r \over r_{max}} \tag 3$$
I get the integral in the form of expression $(1)$:
$$ \int_0^1 \rho J_0(R_n\rho)J_0(R_m\rho)dr={J_1^2(R_n)\over 2} \tag 4 $$
However, I am not sure if this is correct. I have a feeling that the solution to $(2)$ is actually:
$$ \int_0^{r_{max}} {r \over r_{max}} J_0\Bigg({R_nr \over r_{max}}\Bigg)J_0\Bigg({R_nr \over r_{max}}\Bigg)dr= {r_{max}J_1^2(R_n)\over 2} \tag 5 $$
because of the limits of the integral in expression $(2)$. So my question is, is the integral equation $(2)$ really equal to ${J_1^2(R_n)\over 2}$?