Let W be a standard Brownian motion with W 0 = 0 and let b : R + → R be a continuous function with b(0) > 0. The first passage time (from below) is then defined asIt is well-known that the distribution F of τ satisfies a set of Fredholm equations of the first kind, which is used, for example, as a starting point for numerical approaches. For this, it is fundamental that the Fredholm equations have a unique solution. In this article, we prove this in a general setting using analytical methods.