Abstract

Abstract
In this paper, we have applied Weierstrass and Hadamard factorization theorems to obtain the product representation of sin⁡z and cos⁡z and compared the suitability of the two theorems in their applications. It was found that the main challenge in applying Weierstrass theorem is in the identification of the series that results in the course of determining the unknown arbitrary entire function in the theorem. In the case of Hadamard’s theorem the unknown function is a polynomial function of degree at most the order of the entire function whose product representation is being sought. This additional information about the unknown function in the case of Hadamard’s theorem made its determination extremely easy. Hadamards theorem is therefore preferable when factorizing entire functions of finite order. However, Weierstrass theorem is a more general theorem than that of Hadamard. In course of application of Hadamard’s theorem, we also found the maximum modulus of sin⁡z and cos⁡z in the disc |z|≤R and the points at which these maximum occurs. It was also found that the maximum modulus of these functions tends to infinity with R as expected of entire functions. In addition, the two functions were both found to be of order and genus 1, respectively. Further research in this field should focus on applying the two theorems in obtaining the product representation of other entire functions such as sinh⁡z and cosh⁡z and making similar comparisons.

Key words: Keywords and Phrases: Entire Function, Infinite Product, Convergence of Infinite Product, Zeros of entire Functions, Weierstrass theorem, Maximum Modulus of a Function, Order of an entire Function, Hadamard theorem.