The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $mathsf{GL}(N;mathbb{C}),$ in the sense of $ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $bar{Sigma}_{t},$ which is strictly positive and real analytic on $Sigma_{t}$. This density has a simple form in polar coordinates: [ W_{t}(r,theta)=frac{1}{r^{2}}w_{t}(theta), ] where $w_{t}$ is an analytic function determined by the geometry of the region $Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a shadow of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigners semicircle law and Ginibres circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.
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