In this note, we solve the Loewner equation in the upper half-plane with
forcing function xi(t), for the cases in which xi(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 xi(t)=2*(kappa*t)^1/2, the trace is a
straight line set at an angle to the real axis. If xi(t)=2*(kappa*(1-t))^1/2,
the behavior of the trace as t approaches 1 depends on the coefficient kappa.
Our calculations give an explicit solution in which for kappa<4 the trace
spirals into a point in the upper half-plane, while for kappa>4 it intersects
the real axis. We also show that for kappa=9/2 the trace becomes a half-circle.
The third case with forcing xi(t)=t gives a trace that moves outward to
infinity, but stays within fixed distance from the real axis. We also solve
explicitly a more general version of the evolution equation, in which xi(t) is
a superposition of the values +1 and -1.