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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 ?

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Yes, it is. Just define $\sin z$ as$$z-\frac{z^3}{3!}+\frac{z^5}{5!}-\cdots$$It is an entire functions and$$\sin z=0\iff z\in\pi\mathbb{Z}.$$

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