A new class of the entire function of order one: a case study

Abstract

In this article, a new class of the entire function of order one, expressed by the series and product representations with the real positive coefficients and complex zeros, is investigated for the first time. The entire function on the critical line deduces an even entire function of order one. It is proved that the real part of the complex zeros is equal to the critical line. An equivalent representation theorem is obtained to set up the sufficient conditions for the critical line for the entire function. As a typical example, the critical line for the special hyperbolic cosine function obtained by the present theorem agrees with the result of Euler. We also discover the new products of the hyperbolic cosine and sinc functions.