I know that an entire function is one which can be differentiated on the entire complex plane, and I believe that a zero of an entire function is the z where f(z)=0.
I was tring to think of example of the following
An entire function with no zero's: I thought $e^z$ would be suitable as $e^z\neq 0\forall z$
An entire function with one zero : for this I chose $z$ as it is only zero at zero
An entire function with infinite zero's : I thought maybe sin(z) would work here but is sin(z) well defined ? I'm not so sure as $sin(z)=sin(r(cos(\theta) + isin(\theta))$ doesn't seem like a valid expression ( although I could be wrong )
If sin(z) is not well defined what is an example of an entire function with infinite zero's ?