with the product on the left converging if and only if the sum on the right does. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products.
The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète and Wallis:
Product representations of functions
One important result concerning infinite products is that every function f(z) which is entire, i.e. holomorphic over the entire complex plane, can be factored into an infinite product of entire functions each with at most a single zero. In general, if f has a zero of order m at the origin and has other complex zeros at u_{1}, u_{2}, u_{3}, ... (listed with multiplicities equal to their orders) then
where λn are positive integers that can be chosen to make the series converge, and φ(z) is some uniquely determined analytic function (which means the term before the product will have no zeros in the complex plane). The above factorization is not unique, since it depends on the choice of λns, and is not especially elegant. For most functions, though, there will be some minimum positivie integer p such that λn = p gives a convergent product, called the canonical product representation, and in the even that p = 1, this takes the form
This can be regarded as a generalization of the Fundamental Theorem of Algebra, since for polynomials the product becomes finite and φ(z) is constant. Aside from these, the following representations are of special note:
Sine function  sin πz = πz Π (1  z^{2}/n^{2})  Euler  Wallis' formula for π is a special case of this. 
Gamma function  1 / Γ(z) = ze^{γz} Π (1 + z/n) e^{z/n}  Schlömilch 
Riemann zeta function  ζ(z) = Π 1/(1  p_{n}^{z})  Euler  Here p_{n} denotes the sequence of prime numbers. 
Note the last of these is not a product representation of the same sort discussed above, as ζ is not entire.
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