Bessel functions multipied with $\sqrt{\rho}$
$$\sqrt\rho J_\nu(x_{\nu n}\frac{\rho}{a})$$
have orthogonality on $[0, a]$. $x_{\nu n}$ is the nth root of $J_\nu(x)$
$$\int^a_0 \rho J_\nu \left( x_{\nu n'}\frac{\rho}{a} \right)J_\nu \left( x_{\nu n}\frac{\rho}{a} \right)\, d\rho = \frac{a^2}{2}[J_{\nu+1}(x_{\nu n})]^2\delta_{n'n} \tag{1}$$
$\delta$ is Kronecker delta.
Classical Electrodynamics Third Edition mentions because Bessel functions form a complete orthogonal system, any abtriary functions can be represent as
$$f(\rho) = \sum^\infty_{n = 1}A_{\nu n}J_\nu\left( x_{\nu n} \frac{\rho}{a} \right) \tag{2}$$
The orthogonity we get from (1) is $A_{\nu n} \sqrt{\rho} J_\nu\left( x_{\nu n} \frac{\rho}{a} \right)$. Why the basises are $A_{\nu n}J_\nu\left( x_{\nu n} \frac{\rho}{a} \right)$ instead of $A_{\nu n} \sqrt{\rho} J_\nu\left( x_{\nu n} \frac{\rho}{a} \right)$ in (2)?