Entire Functions of Order One and Infinite Type

Abstract

In this paper we first prove an auxiliary result that an entire function of order one and infinite type must have infinitely many non-zero zeros. We then give an explicit canonical representation for those functions. We apply the representation to prove a result and its converse about entire functions of order one and infinite type. Next, we mention a few interesting examples of entire functions of order one and infinite type. Finally, we formulate and disprove a conjecture which serves as an analogue to the Paley-Wiener Theorem for entire functions of order one and infinite type.