Infinite product

In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product

\prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots

is defined to be the limit of the partial products a1a2...an as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence an as n increases without bound must be 1, while the converse is in general not true.

The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product):

\frac{2}{\pi} = \frac{ \sqrt{2} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2}} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2 + \sqrt{2}}} }{ 2 } \cdots
\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \prod_{n=1}^{\infty} \left( \frac{ 4 \cdot n^2 }{ 4 \cdot n^2 - 1 } \right).

Convergence criteria

The product of positive real numbers

\prod_{n=1}^{\infty} a_n

converges to a nonzero real number if and only if the sum

\sum_{n=1}^{\infty} \ln(a_n)

converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. The same criterion applies to products of arbitrary complex numbers (including negative reals) if the logarithm is understood as a fixed branch of logarithm which satisfies ln(1) = 0, with the proviso that the infinite product diverges when infinitely many an fall outside the domain of ln, whereas finitely many such an can be ignored in the sum.

For products of reals in which each a_n\ge1, written as, for instance, a_n=1+p_n, where p_n\ge 0, the bounds

1+\sum_{n=1}^{N} p_n \le \prod_{n=1}^{N} \left( 1 + p_n \right) \le \exp \left( \sum_{n=1}^{N}p_n \right)

show that the infinite product converges precisely if the infinite sum of the pn converges. This relies on the Monotone convergence theorem. More generally, the convergence of \prod_{n=1}^\infty(1+p_n) is equivalent to the convergence of \sum_{n=1}^\infty p_n if pn are real or complex numbers such that \sum_{n=1}^\infty|p_n|^2<+\infty, since \ln(1+x)=x+O(x^2) in a neighbourhood of 0.

If the series pn diverges to zero, then the sequence of partial products of the pn converges to zero as a sequence. The infinite product is said to diverge to zero.[1]

Product representations of functions

One important result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions, each with at most a single root. In general, if f has a root of order m at the origin and has other complex roots at u1, u2, u3, ... (listed with multiplicities equal to their orders), then

f(z) = z^m e^{\phi(z)} \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n} \right) \exp \left\lbrace \frac{z}{u_n} + \frac{1}{2}\left(\frac{z}{u_n}\right)^2 + \cdots + \frac{1}{\lambda_n} \left(\frac{z}{u_n}\right)^{\lambda_n} \right\rbrace

where λn are non-negative integers that can be chosen to make the product converge, and φ(z) is some uniquely determined analytic function (which means the term before the product will have no roots in the complex plane). The above factorization is not unique, since it depends on the choice of values for λn, and is not especially elegant. However, for most functions, there will be some minimum non-negative integer p such that λn = p gives a convergent product, called the canonical product representation. This p is called the rank of the canonical product. In the event that p = 0, this takes the form

f(z) = z^m e^{\phi(z)} \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n}\right).

This can be regarded as a generalization of the Fundamental Theorem of Algebra, since the product becomes finite and φ(z) is constant for polynomials.

In addition to these examples, the following representations are of special note:

Simple pole \frac{c}{c- z}=\prod_{n=1}^\infty e^{\frac{1}{n}\,\left( \frac{z}{c}\right)^n}
Sinc function \textrm{sinc}(\pi z) =\prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right) This is due to Euler. Wallis' formula for π is a special case of this.
Gamma function \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-\frac{z}{n}} Schlömilch
Weierstrass sigma function \sigma(z) = z\prod_{\omega \in \Lambda_{*}} \left(1-\frac{z}{\omega}\right)e^{\frac{z^2}{2\omega^2}+\frac{z}{\omega}} Here \Lambda_{*} is the lattice without the origin.
Q-Pochhammer symbol (z;q)_\infty = \prod_{n=0}^\infty (1-zq^n) Widely used in q-analog theory. The Euler function is a special case.
Ramanujan theta function \begin{align} f(a,b) &=\sum_{n=-\infty}^\infty a^{\frac{n(n+1)}{2}} b^{\frac{n(n-1)}{2}} \\ &= \prod_{n=0}^\infty (1+a^{n+1}b^n)(1+a^nb^{n+1})(1-a^{n+1}b^{n+1}) \end{align} An expression of the Jacobi triple product, also used in the expression of the Jacobi theta function
Riemann zeta function \zeta(z) = \prod_{n=1}^{\infty} \frac{1}{1 - p_n^{-z}} Here pn denotes the sequence of prime numbers. This is a special case of the Euler product.

Note that the last of these is not a product representation of the same sort discussed above, as ζ is not entire. Rather, the above product representation of ζ(z) converges precisely for Re(z) > 1, where it is an analytic function. By techniques of analytic continuation this function can be extended uniquely to an analytic function (still called ζ(z)) on the whole complex plane except for the point z=1, where it has a simple pole.

See also

References

  1. ↑ Jeffreys, Harold; Jeffreys, Bertha Swirles (1999). Methods of Mathematical Physics. Cambridge Mathematical Library (3rd revised ed.). Cambridge University Press. p. 52. ISBN 1107393671.

External links

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