  Consider a Hermitian matrix model under an external potential with spiked
external source. When the external source is of rank one, we compute the
limiting distribution of the largest eigenvalue for general, regular, analytic
potential for all values of the external source. There is a transitional
phenomenon, which is universal for convex potentials. However, for non-convex
potentials, new types of transition may occur. The higher rank external source
is analyzed in the subsequent paper.
