We study the dynamics of "finger" formation in Laplacian growth without sur
face tension in a channel geometry (the Saffman-Taylor problem). We present
a pedagogical derivation of the dynamics of the conformal map from a strip
in the complex plane to the physical channel. In doing so we pay attention
to the boundary conditions (no flux rather than periodic) and derive a fie
ld equation of motion for the conformal map. We first consider an explicit
analytic class of conformal maps that form a basis for solutions in infinit
ely long channels. characterized by meromorphic derivatives. The great bulk
of these solutions can lose conformality due to finite time singularities.
By considerations of the nature of the analyticity of these solutions, we
show that those solutions which are free of such singularities inevitably r
esult in a single asymptotic "finger" whose width is determined by initial
conditions. This is in contradiction with the experimental results that ind
icate selection of a finger of width 1/2, In the last part of this paper we
show that such a solution might be determined by the boundary conditions o
f a finite body of fluid, e.g. finiteness can lead to pattern selection.