# An example of an entire function, which is not polynomial, that has only one zero.

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.

• Do you know a non-polynomial entire function with no zero ? A polynomial with only one zero ? The product of both ? – reuns Dec 15 '19 at 22:18
• 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.. – variableXYZ Dec 15 '19 at 22:29
• @variableXYZ On what basis would you call $ze^z$ a polynomial? – Kevin Carlson Dec 15 '19 at 22:35
• @variableXYZ $ze^z$ is not a polynomial..it is a pointwise limit of a sequence of polynomials though – Marios Gretsas Dec 15 '19 at 22:36

You can take$$\begin{array}{ccc}\mathbb C&\longrightarrow&\mathbb C\\z&\mapsto&ze^z.\end{array}$$