I'm looking for nonconstant entire functions $f(z)$ such that $f(z)\neq 0$ for any $z$.
More specifically I'm looking for nontrivial cases.
So $\exp(z),\exp(z^2),...$ is not what I am looking for.
I am aware that every solution is essentially equal to $\exp(g(z))$ for some entire $g$, but that does not mean every answer needs to be in that form.
For instance f is never zero and f is given by An integral transform; so COMPOSITION of Some g and exp is not explicitly given. Or g has no closed form.
Hope this clarifies the confusion.
( in case a definition for entire f does not hold/converge everywhere you may use analytic continuation )
Additionally I wonder if such a function can satisfy a simple functional equation of any kind.
Assume f(z)= 2^z + 3^z - 7^z is never equal to 0.
Then ln(2^z + 3^z - 7^z) is entire and equal to g(z). F(z) = exp(g(z)).
Hope this clarifies.
Maybe this related question makes it clear.
This question is a generalisation of it. Hope it finally helps before this gets closed.