i have to add some conditions and ideas :

if f(z) = f(g(z)) then we have product terms of the form 1 + b_n g(z)^n.

those product terms have different zero's despite f(z) = f(g(z)).

also it seems if f ' (z) = 0 our product form does not work.

so i propose the following condition.

the product form of f(z) converges to the correct value in Q if f(z) =/= 0 and f(z) is univalent in Q.

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it would be nice to consider product forms of f^[n](z).

is the expression for f(z) = e^z - 1 and it would be nice to have a similar looking product form ...

if f(z) = f(g(z)) then we have product terms of the form 1 + b_n g(z)^n.

those product terms have different zero's despite f(z) = f(g(z)).

also it seems if f ' (z) = 0 our product form does not work.

so i propose the following condition.

the product form of f(z) converges to the correct value in Q if f(z) =/= 0 and f(z) is univalent in Q.

***

it would be nice to consider product forms of f^[n](z).

is the expression for f(z) = e^z - 1 and it would be nice to have a similar looking product form ...