I was reading the section of Lang's Complex Analysis about Weierstrass products, and came across this example:
I don't understand how the textbook got these results. So far in this section, I have read that
If f is an entire function without zeros, there exists an entire function h such that $f(z) = e^{h(z)}$
If $|z| \leq 1/2$, then $| \log E_n(z) |\leq 2|z|^n$
The product $\Pi_{n=1}^\infty E_n(z,z_n) = \Pi_{n=1}^\infty (1-\frac{z}{z_n})e^{P_n(z/z_n)} $ converges uniformly and absolutely on every disc $|z| \leq R$ and defines an entire function with zeros at the points of the sequence $\{z_n\}$, and no other zeros
How did the author get to the conclusions in the example? I included these theorems since I assume they are used. If someone could provide clues or break it down for me, it would be much appreciated. Also any links to useful resources that explain Weierstrass products would be a help.