Weierstrassfaktorisasjon

Weierstrass factorization is a method in complex analysis used to express entire functions as products of their zeros and a canonical factor. It is named after the German mathematician Karl Weierstrass. The factorization is particularly useful for meromorphic functions, which are functions that are holomorphic (analytic) on a domain except for isolated singularities.

The Weierstrass factorization theorem states that any entire function f(z) can be written as:

f(z) = e^g(z) * ∏ (1 - z/z_n) * ∏ (1 - z/z_n)^-1

where g(z) is an entire function, z_n are the zeros of f(z), and z_n are the poles of f(z). The first product is over all zeros of f(z), and the second product is over all poles of f(z). The term (1 - z/z_n) is raised to the power of the order of the zero at z_n, and (1 - z/z_n)^-1 is raised to the power of the order of the pole at z_n.

The Weierstrass factorization is a powerful tool in complex analysis, as it allows for the study of the behavior of functions at their zeros and poles. It is also used in the study of special functions, such as the gamma function and the Riemann zeta function.