For a sequence
of numbers a1
, ... we define the infinite product
... to be the limit
of the partial products a1a2
goes to infinity. When the limit exists we say the product converges
and when it does not we say that it diverges
, except that the product is still considered to be divergent when the limit is zero, in order to get results analogous to those for infinite sums
. If the product converges, then the limit of the sequence an
goes to infinity must be 1 (the converse is in general not true). Therefore, the logarithm
will be defined for all but finitely many n
, and for those we have
- log Π an = ∑ log an
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 &pi, such as the following two products, respectively by Viète and Wallis:
Product representations of functions
- 2/π = (√2 / 2)(√(2 + √2) / 2)(√(2 + √(2 + √2)) / 2)...
- π/2 = (2/1)(2/3)(4/3)(4/5)(6/5)(6/7)(8/7)(8/9)...
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 u1, u2, u3, ... (listed with multiplicities equal to their orders) then
- f(z) = zmeφ(z) Π (1 - z/un) exp[z/un + (z/un)2/2 + ... + (z/un)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 λn
s, and is not especially elegant. For most functions, though, there will be some minimum positivie integer p
such that λn
gives a convergent product, called the canonical product representation, and in the even that p
= 1, this takes the form
- f(z) = zmeφ(z) Π (1 - z/un)
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:
Note the last of these is not a product representation of the same sort discussed above, as ζ is not entire.