  We study the dynamics of "finger" formation in Laplacian growth without
surface tension in a channel geometry (the Saffman-Taylor problem). Carefully
determining the role of boundary geometry, we construct field equations of
motion, these central to the analytic power we can here exercise. We consider
an explicit analytic class of maps to the physical space, a basis of solutions
for infinite fluid in an infinitely long channel, characterized by meromorphic
derivatives. We verify that these maps never lose analyticity in the course of
temporal evolution, thus justifying the underlying machinery. However, the
great bulk of these solutions can lose conformality in time, this the
circumstance of finite-time singularities. By considerations of the nature of
the analyticity of all these solutions, we show that those free of such
singularities inevitably result in a {\em single} asymptotic "finger". This is
purely nonlinear behavior: the very early "finger" actually already has a
waist, this having signalled the end of any linear regime. The single "finger"
has nevertheless an arbitrary width determined by initial conditions. This is
in contradiction with the experimental results that indicate selection of a
finger of width 1/2. In the last part of this paper we motivate that such a
solution can be determined by the boundary conditions when the fluid is finite.
This is a strong signal that {\em finiteness} is determinative of pattern
selection.
