In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. The theorem, which is named for Karl Weierstrass, is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. A generalization of the theorem extends it to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function.
Motivation It is clear that any finite set { c n } {\displaystyle \{c_{n}\}} of points in the complex plane has an associated polynomial p ( z ) = ∏ n ( z − c n ) {\textstyle p(z)=\prod _{n}(z-c_{n})} whose zeroes are precisely at the points of that set. The converse is a consequence of the fundamental theorem of algebra: any polynomial function p ( z ) {\displaystyle p(z)} in the complex plane has a factorization
p ( z ) = a ∏ n ( z − c n ) , {\textstyle p(z)=a\prod _{n}(z-c_{n}),} where a is a non-zero constant and { c n } {\displaystyle \{c_{n}\}} is the set of zeroes of p ( z ) {\displaystyle p(z)} . The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of additional terms in the product is demonstrated when one considers ∏ n ( z − c n ) {\textstyle \prod _{n}(z-c_{n})} where the sequence { c n } {\displaystyle \{c_{n}\}} is not finite. It can never define an entire function, because the infinite product does not converge. Thus one cannot, in general, define an entire function from a sequence of prescribed zeroes or represent an entire function by its zeroes using the expressions yielded by the fundamental theorem of algebra. Instead, the theorem replaces these with other factors. A necessary condition for convergence of the infinite product in question is that for each z {\displaystyle z} , the factors replacing ( z − c n ) {\displaystyle (z-c_{n})} must approach 1 as n → ∞ {\displaystyle n\to \infty } . So it stands to reason that one should seek factor functions that could be 0 at a prescribed point, yet remain near 1 when not at that point, and furthermore introduce no more zeroes than those prescribed. Weierstrass' elementary factors have these properties and serve the same purpose as the factors ( z − c n ) {\displaystyle (z-c_{n})} above.
Elementary factors Consider the functions of the form exp ( − z n + 1 n + 1 ) {\textstyle \exp \left(-{\tfrac {z^{n+1}}{n+1}}\right)} for n ∈ N {\displaystyle n\in \mathbb {N} } . At z = 0 {\displaystyle z=0} , they evaluate to 1 {\displaystyle 1} and have a flat slope at order up to n {\displaystyle n} . Right after z = 1 {\displaystyle z=1} , they sharply fall to some small positive value. In contrast, consider the function 1 − z {\displaystyle 1-z} which has no flat slope but, at z = 1 {\displaystyle z=1} , evaluates to exactly zero. Also note that for |z| < 1,
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