Weierstrass factorization theorem: Difference between revisions

Revision as of 10:04, 15 February 2025

Revision as of 10:07, 15 February 2025

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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 <math display="inline">\prod_n (z-c_n)</math> where the sequence <math>\{c_n\}</math> is not [[finite set|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.

A necessary condition for convergence of the infinite product in question is that for each z, the factors replacing <math> (z-c_n) </math> must approach 1 as <math>n\to\infty</math>. So it stands to reason that one should seek a functionseries of 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 <math> (z-c_n) </math> above.