In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, 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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. He was born in Ostenfelde, Westphalia (today Germany) and died in Berlin, Germany. ... In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ...

A second form extended to meromorphic functions allows one to consider a given meromorphic function as a product of three factors: the function's poles, zeroes, and an associated non-zero holomorphic function. In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles fore the function. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles fore the function. ...

Motivation

The consequences of the fundamental theorem of algebra are twofold⁴. Firstly, any finite sequence,{c_n}, in the complex plane has an associated polynomial that has zeroes precisely at the points of that sequence: In mathematics, the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n roots (zeroes), counted with multiplicity. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... 0 (zero), alternatively called aught or nought, is both a number and a numeral. ... This is a page about mathematics. ...

Secondly, any polynomial function in the complex plane, p(z), has a factorization In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ...

where a is a non-zero constant and c_n are the zeroes of p.

The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of extra machinery is demonstrated when one considers whether the product In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ...

defines a entire function if the sequence, {c_n}, is not finite. The answer is 'not always', because the now-infinite product may not converge in the (entire) plane. 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. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ...

A heuristic condition for convergence of the infinite product in question is: each factor, (z − c_n), should be "near" 1. So it stands to reason that one should seek a function 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. Enter the genius of Weierstrass' elementary factors. These factors serve the same purpose as the factors, (z − c_n), above.

The elementary factors

These are also referred to as primary factors⁵.

For , define the elementary factors¹:

Their utility lies in the following lemma¹:

Lemma (15.8, Rudin) for

The two forms of the theorem

Sequences define holomorphic functions

Sometimes called the Weierstrass Theorem ³

If is a sequence such that:

1. as
2. there is a sequence, , such that:

Then there exists an entire function that has (only) zeroes at every point of {z_i}; in particular, P is such a function¹:

• The theorem generalizes to: sequences in open subsets (and hence regions) of the Riemann sphere have associated functions that are holomorphic in those subsets and have zeroes at the points of the sequence. ¹
• Note also that the case given by the FTA is incorporated here. If the sequence, {z_i} is finite then suffices for convergence in condition 2, and we obtain: .

This is a page about mathematics. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... A region can be any area that has some unifying feature. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Holomorphic functions can be factored

Sometimes called the Weierstrass Product/Factor/Factorization theorem.[3] Sometimes called the Hadamard Factorization theorem; for example c.f. ⁵.

If f is a function holomorphic in a region, Ω, with zeroes at every point of then there exists an entire function g, and a sequence such that:

• • There is a a unique factorization if Π_iz_i is convergent¹.
  • The theorem may be generalized to the space of meromorphic functions, in which case, the factorization is unique. Let f be a meromorphic function and be the zeroes and poles of the function, respectively; then: .

A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ...

References

• Note 1: Rudin, W, Real and Complex Analysis, 3rd Ed, Mc Graw Hill, Boston, pp 301 - 304, 1987
• Note 2: Eric W. Weisstein. "Weierstrass Product Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/WeierstrassProductTheorem.html
• Note 3: Eric W. Weisstein. "Weierstrass's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/WeierstrasssTheorem.html
• Note 4: Knopp, K. "Weierstrass's Factor-Theorem", §1 in "Theory of Functions" Part II. New York: Dover, pp. 1-7, 1996.
• Note 5: Boas, R. P., "Entire Functions", Academic Press Inc., New York, 1954, chapter 2.