A note on the entire functions: theorems, properties and examples

Abstract

In this work, we summarized several fundamental theorems of the entire functions and explored how their zeros determine those functions. The background of this work is based on Elias M. Stein and Rami Shakarachi, and their lectures about functions that are holomorphic in the whole complex plane that once were taught at Princeton University. First, we introduced Jensen’s formula and gave detailed proof. This relates the values of a meromorphic function inside a disk with its boundary values on the circumference and with its zeros and poles. Next, we studied the proof of Weierstrass infinite products and the definition of canonical factors. Last, Hadamard’s factorization theorem and a few main lemmas were introduced. We showed the proof of the Hadamard factorization theorem and solved its several applications through using examples. Hadmard’s theory is well demonstrated in this article, and we showed the rigor and scientific validity of the theory through examples.