the Weierstrass factorization theorem
/ VY-er-shtrahss /
A polynomial is completely captured by its roots: p(z) = c (z - r_1) ... (z - r_n). The Weierstrass factorization theorem is the audacious claim that EVERY entire function — every function holomorphic on the whole plane — admits an analogous factored form built from its zeros, even when there are infinitely many of them. It is the fundamental theorem of algebra grown up to the transcendental case.
Here is the statement. Let f be entire with a zero of order m at the origin (m could be 0) and other zeros at points a_1, a_2, ... listed with multiplicity, with |a_n| -> infinity. Then f(z) = z^m e^(g(z)) times the product over n of E_(p_n)(z/a_n), where g is some entire function and the E_(p_n) are Weierstrass elementary factors with degrees p_n chosen large enough to force the product to converge. Read it piece by piece: z^m handles the zero at the origin; the convergent product over the elementary factors plants a zero of the correct multiplicity at each a_n and nowhere else; and the unit factor e^(g(z)) — entire and never zero — absorbs everything about f that the zeros cannot see. Two entire functions with the same zeros differ exactly by such a non-vanishing factor e^g.
The honest content is twofold. Existence: the elementary factors guarantee a convergent product with precisely the prescribed zeros, so such a factorization always exists. Non-uniqueness: the theorem does NOT pin down g — the zeros determine f only up to multiplication by e^g, because you can multiply by any never-zero entire function without disturbing a single zero. Pinning g down requires extra information (like the growth rate of f), which is the business of the Hadamard factorization theorem, not this one. Weierstrass tells you the zeros can always be realized; it does not tell you they determine the function.
The entire function sin(pi z) has simple zeros at every integer n. Its Weierstrass factorization comes out as sin(pi z) = pi z times the product over n != 0 of E_1(z/n) = pi z product over n >= 1 of (1 - z^2/n^2). Here m = 1 (a zero at the origin), the degree p_n = 1 suffices because the integers grow linearly, and the e^g factor turns out to be just the constant giving the prefactor pi — no exponential is needed.
sin(pi z) factored over its integer zeros — the prototype Weierstrass product, with elementary factors of degree 1.
The theorem provides existence and a shape, not uniqueness: the factor e^(g(z)) is genuinely free, so the zeros alone never determine an entire function. Forgetting this, and reading the factorization as 'the' formula, is a common error — two functions can share all zeros and still differ wildly.