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According to wikipedia,

"...the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes."

Yet the exponential function is an entire function with no zeros, is there a implicit stipulation in this theorem that entire functions must have at least one zero in order for the theorem to apply?

Note: Maybe this is stupidly obvious, but I wan't to make sure there isn't a more advanced point of view which resolves this by considering points whose real part approaches negative infinity, or something of the like.

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In the Weierstrass factorization theorem, the product is allowed to have a factor of $e^{g(z)}$ outside, so since the exponential function has no zeros, it is just represented by $e^z$.

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