Why the factor $e^{g(z)}$ in the Weierstrass factorization theorem? The Weierstrass factorization theorem is usually stated as such (quote from Wikipedia):



Let $f$ be an entire function, and let $\{a_n\}$ be the non-zero zeros of $f$ repeated according to multiplicity; suppose also that $f$ has a zero at $z = 0$ of order $m ≥ 0$ (...). Then there exists an entire function $g$ and a sequence of integers $\{p_{n}\}$ such that

$$f(z)=z^me^{g(z)}\prod_{n=1}^\infty E_{p_n}\left(\frac{z}{a_n}\right).$$



It seems to me that this combines two different theorems: that $f$ can be written as
$$f(z)=z^mh(z)\prod_{n=1}^\infty E_{p_n}\left(\frac{z}{a_n}\right)$$
for some entire function $h$ with no zeros, and that any entire function $h$ with no zeros can be written as $h(z)=e^{g(z)}$ for some entire function $g$. The latter is an interesting nontrivial statement in its own right, and, as far as I can tell, its proof is not connected with the rest of the proof of the Weierstrass factorization theorem.

So, what is the mathematical or historical reason for writing the theorem with $e^{g(z)}$ instead of $h(z)$?
I believe the reason is linked to the Argumente Principle. But the main property of writing e^h(z) is the fact that that function is never zero.
I believe the reason is linked to the Argumente Principle. But the main property of writing e^h(z) is the fact that that function is never zero.