While there are a number of techniques for evaluating partition functions describing helix-coil transitions in infinite biopolymers, few exact techniques exist for the finite polymer problem. In this work, the sequence generating function formalism is extended to allow the treatment of finite length biopolymers. This is achieved by representing the partition function generating function as an expansion that uses a descending series rather than an ascending series. In this formalism the partition function is directly determined by a contour integral that can be evaluated exactly by the calculus of residues. This exact evaluation has advantages over previously used steepest descent methods as it allows finite biopolymers to be treated. With this technique, order-disorder transitions in a nucleic acid double helix model are reexamined. The problem of the perfectly matched double helix is treated exactly. The finite case of this model requires that the roots of a trinomial equation be obtained. This equation is solved and it is shown that each root differs only by a phase factor. Thus, the roots are distributed about a circle in the complex plane. Yang-Lee theory is used to explore conditions under which first-order phase transitions will occur in these systems.