Infinite productFor a sequence of numbers a1, a2, a3, ... we define the infinite product Π an = a1a2a3... to be the limit of the partial products a1a2...an as n 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 as n goes to infinity must be 1 (the converse is in general not true). Therefore, the logarithm log an will be defined for all but finitely many n, and for those we have
- log Π an = ∑ log an
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:
- 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]
- f(z) = zmeφ(z) Π (1 - z/un)
|Sine function||sin πz = πz Π (1 - z2/n2)||Euler - Wallis' formula for π is a special case of this.|
|Gamma function||1 / Γ(z) = zeγz Π (1 + z/n) e-z/n||Schlömilch|
|Riemann zeta function||ζ(z) = Π 1/(1 - pn-z)||Euler - Here pn 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.