From an explicit formula for the joint density of the radial part and the winding number of a planar Brownian motion, we obtain asymptotic expansions (as t tends to infinity) for the density of the winding number. In particular, this expansion yields corrections of any order (in inverse powers of log(t)) to Spitzer's law (Trans. Am. Math. Soc, 1958) and to a local limit theorem proved by Delbaen, Kowalski and Nikehbali (Int. Math. Research Notices, 2014).