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Weierstrass factorization theorem

The Weierstrass factorization theorem asserts that entire functions can be represented by a (possibly infinite) product involving their zeroes.

 

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Explanation

You can dissolve every complex polynomial in one variable z into linear factors

a₀ + a₁z + a₂z² + ··· + a_nzⁿ = a_n(z − b₁)(z − b₂) ··· (z − b_n)

 

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Example 1

The zeroes of the function

are integer multiples of π, so the points x = n · π for n = ±1,  ±2,  ±3, ... . The theorem gives then

 

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History The factorization theorem was described by the German mathematician Karl Weierstrass in 1876.

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