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.


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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