Weierstrass factorization theorem

The two forms of the Weierstrass Factorization theorem can be thought of as the extensions of the above theory to the space of entire functions. The necessity of the extra machinery is demonstrated when one considers whether the product defines a entire function if the sequence, , 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.

Note that a heuristic condition for convergence of the infinite product in question is: each factor, , 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, , above.

The elementary factors

For , define the elementary factors:

Their utility lies in the following lemma:

Lemma (15.8, Rudin) for

The two forms of the theorem

Sequences define holomorphic functions

If
is a sequence such that:

• It is worthy of note that 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.
• Note also that the case given by the FTA is incorporated here. If the sequence, is finite then suffices for convergence in condition 2, and we obtain: .
  

  Holomorphic functions can be factored

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

• • There is a a unique factorization if is convergent.
  • *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: .
    

    References

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