I was studying previous complex analysis exams and I came across the following question:
Use a result obtained from the residue theorem to justify that following infinite product is representation is valid $$ e^z-1=z e^{\frac{z}{2}} \prod_{n=1}^{\infty}\left(1+\frac{z^2}{4 n^2 \pi^2}\right) $$
I saw some solutions using a product expansion for $\frac{\sin(\pi z)}{\pi z}$ that didn't use the residue theorem. Our instructor also presented the formula without proof $$ f(z)=f(0) e^{\frac{f^{\prime}(0)}{f(0)}}\prod_{k=1}^{\infty}\left(1-\frac{z}{a_k}\right) e^{\frac{z}{a_k}}. $$ Where $f(z)$ is an entire function with zeroes of order $1$ on $a_k$. I would like to know a reference, or a proof, for the above formula and how it is related to the residue theorem.