Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms

This is the sixth part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part VI), we construct the appropriate natural associativity isomorphisms between triple tensor product functors. In fact, we establish a "logarithmic operator product expansion" theorem for logarithmic intertwining operators. In this part, a great deal of analytic reasoning is needed; the statements of the main theorems themselves involve convergence assertions.