Weierstrass's Theorem

There are at least two theorems known as Weierstrass's theorem. The first states that the only hypercomplex number systems with commutative multiplication and addition are the algebra with one unit such that I=I^2 and the Gaussian integers.

In harmonic analysis, let U subset= C be any open set, and let a_1, a_2, ..., be a finite or infinite sequence in U (possibly with repetitions) that has no accumulation point in U. There exists an analytic function f on U whose zero set is precisely {a_j} (Krantz 1999, p. 111). This is also sometimes known as the Weierstrass product theorem.

See also

Gaussian Integer, Hypercomplex Number, Peirce's Theorem, Weierstrass Product Theorem

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Krantz, S. G. "Weierstrass's Theorem" §8.3.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 111, 1999.

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Weierstrass's Theorem

Cite this as:

Weisstein, Eric W. "Weierstrass's Theorem." From MathWorld--A Wolfram Web Resource.

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