proving than an infinite product defines an entire function

Question

Consider the infinite product $$F(z)=\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$ How can i prove that $F$ is entire? Can i write $F$ as a Weierstrass product $\prod E(\frac{z}{z_j},p_j)$, for suitable zeros $z_j's$ and integers $p_j's$?

Answer 1

A product $\prod_{n=1}^\infty (1-f_n(z))$ converges absolutely and uniformly if $\sum_{n=1}^\infty f_n(z)$ does.

For your example $|f_n(z)| = |e^{-2\pi n} e^{-iz}| = e^{-2\pi n}e^y$ (with $z=x+iy$). Let $\Omega_k = \{ z = x+iy : |y| < k \}$. By Weierstrass M-test, the series $\sum_{n=1}^\infty f_n(z)$ converges uniformly on $\Omega_k$, so $F$ is holomorphic on $\Omega_k$, and this holds for every $k$. Hence $F$ is entire.