10,238 research outputs found
A new class of the entire functions: a study of two cases from Euler
In this article an alternative infinite product for a special class of the
entire functions are studied by using some results of the Laguerre-P\'{o}lya
entire functions. The zeros for a class of the special even entire functions
are discussed in detail. It is proved that the infinite product and series
representations for the hyperbolic and trigonometric cosine functions, which
are coming from Euler, are our special cases.Comment: 10 page
A new class of the entire function of order one: a case study
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.Comment: V1. there are 11 pages. V2, there are 20 pages. We consider some new
contents for the entire function
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