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I cannot find an example of an entire function $f$ (i.e. $f: \mathbb{C}\rightarrow\mathbb{C}$), which is not polynomial, that has only one zero. Besides these two, I had to find examples with no zeros and infinitely many, but I found them.

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    $\begingroup$ Do you know a non-polynomial entire function with no zero ? A polynomial with only one zero ? The product of both ? $\endgroup$ – reuns Dec 15 '19 at 22:18
  • $\begingroup$ I took $f(z)=z$ as a polynomial with only one zero and $e^z$ as non-polynomial with no zero? But the product is still polynomial.. $\endgroup$ – variableXYZ Dec 15 '19 at 22:29
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    $\begingroup$ @variableXYZ On what basis would you call $ze^z$ a polynomial? $\endgroup$ – Kevin Carlson Dec 15 '19 at 22:35
  • $\begingroup$ @variableXYZ $ze^z$ is not a polynomial..it is a pointwise limit of a sequence of polynomials though $\endgroup$ – Marios Gretsas Dec 15 '19 at 22:36
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You can take$$\begin{array}{ccc}\mathbb C&\longrightarrow&\mathbb C\\z&\mapsto&ze^z.\end{array}$$

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