Frobenius & Special Functions · Hard problem

Bessel's equation of order $\nu$, $x^2y'' + xy' + (x^2 - \nu^2)y = 0$, has indicial equation $r^2 - \nu^2 = 0$ at its regular singular point $x = 0$. For $\nu = \tfrac12$, compute the positive indicial root. Express your answer as a common fraction.