Exact Solutions for Loewner Evolutions

In this note, we solve the Loewner equation in the upper halfplane with forcing function ξ(t), for the cases in which ξ(t) has a power-law dependence on time with powers 0, 1/2 and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If ξ(t) = 2 √ κt, the trace is a straight line set at an angle to the real axis. If ξ(t) = 2 √ κ(1 − t), as pointed out by Marshall and Rohde [12], the behavior of the trace as t approaches 1 depends on the coefficient κ. Our calculations give an explicit solution in which for κ < 4 the trace spirals into a point in the upper half-plane, while for κ > 4 it intersects the real axis. We also show that for κ = 9/2 the trace becomes