Weierstrass factorization theorem

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes. In addition, every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. The theorem is named after Karl Weierstrass.

A second form of the theorem extends to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's poles and zeroes, and an associated non-zero holomorphic function.

The consequences of the fundamental theorem of algebra are twofold. Firstly, any finite sequence ${\displaystyle \{c_{n}\}}$ in the complex plane has an associated polynomial ${\displaystyle p(z)}$ that has zeroes precisely at the points of that sequence, ${\displaystyle p(z)=\,\prod _{n}(z-c_{n}).}$

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