偶次幂为非拟素的有穷级素整函数的存在性

Existence of Prime Entire Function of Finiite Order Whose Even Powers Are not Pseudo-prime

本文解决了[1]中提出的一个问题,即证明了:存在有穷级的素整函数F(z),使得E(z)~2是非拟素的。同时进一步证明:若F(z)是拟素的超越整函数,而F(z)~2可分解为F^2=f(g),这里f,g为超越整函数,则g(z)=cosα(z),以及F(z)=sinα(z)h(cosα(z)),这里α(z),h(w)是整函数。

The solution to a problem posed in [1] is given.The authors prove that there exists a prime entire function F(z)of finite order such that F(z)~2 is not pseudoprime.It is also proved that if F is a pseudo-prime transcendental entire function,and if F^2 is factorized as F^2=fog,where f and g are both transcendental entire,then g(z)=cosα(z)and F(z)=sin α(z)h(cos(z))with α(z),h(z)being entire.