Weierstrass factorization theorem

The theorem that every nonzero entire function is a canonical product determined by its zeros multiplied by the exponential of an entire function.

Weierstrass primary factor: The factor E_p(z)=(1−z)exp(z+z²/2+⋯+z^p/p), used to control convergence in canonical products. Its exponential correction lets factors encode zeros without making the product diverge.
Holomorphic function: A complex function that is complex differentiable throughout an open domain. Entire functions are holomorphic on the whole complex plane.
Karl Weierstrass: A German mathematician whose work established foundational results in analysis, including the factorization theorem bearing his name. He developed the convergence-controlled products that make the theorem possible.
Gamma function: The meromorphic function extending factorials, satisfying Γ(z+1)=zΓ(z). Its reciprocal has a canonical product whose zeros are the nonpositive integers.
Maximum modulus principle: The principle that a nonconstant holomorphic function cannot attain a local maximum of its modulus inside its domain. It constrains entire functions through their values, whereas factorization organizes them through zeros.
Canonical product: An infinite product built from Weierstrass primary factors and a sequence of zeros. It is the product form that carries the function’s prescribed zeros.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an interior accumulation point agree throughout a connected domain. It helps explain why zero data can strongly constrain an analytic function.
Bernhard Riemann: A German mathematician whose work shaped complex analysis, geometry, and number theory. His approach to analytic functions formed part of the mathematical setting Weierstrass helped formalize.
Sine function: The entire function sin(z), with zeros at integer multiples of π. Its infinite product illustrates how a familiar function is encoded by its zeros.
Liouville's theorem: The theorem that every bounded entire function is constant. It shows how a growth restriction can sharply limit the functions allowed by a zero set.

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