Infinite product

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

${\displaystyle \prod _{n=1}^{\infty }a_{n}=a_{1}a_{2}a_{3}\cdots }$

is defined to be the limit of the partial products a₁a₂...a_n 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 a_n 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):

${\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \;\cdots =\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}}$

${\displaystyle {\frac {\pi }{2}}=\left({\frac {2}{1}}\cdot {\frac {2}{3}}\right)\cdot \left({\frac {4}{3}}\cdot {\frac {4}{5}}\right)\cdot \left({\frac {6}{5}}\cdot {\frac {6}{7}}\right)\cdot \left({\frac {8}{7}}\cdot {\frac {8}{9}}\right)\cdot \;\cdots =\prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right).}$

Convergence criteria

The product of positive real numbers

${\displaystyle \prod _{n=1}^{\infty }a_{n}}$

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

${\displaystyle \sum _{n=1}^{\infty }\log(a_{n})}$

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 ${\displaystyle \ln(1)=0}$, with the proviso that the infinite product diverges when infinitely many a_n fall outside the domain of ${\displaystyle \ln }$, whereas finitely many such a_n can be ignored in the sum.

If we define ${\displaystyle a_{n}=1+p_{n}}$, the bounds

${\displaystyle 1+\sum _{n=1}^{N}p_{n}\leq \prod _{n=1}^{N}\left(1+p_{n}\right)\leq \exp \left(\sum _{n=1}^{N}p_{n}\right)}$

show that the infinite product of a_n converges if the infinite sum of the p_n converges. This relies on the Monotone convergence theorem. We can show the converse by observing that, if ${\displaystyle p_{n}\to 0}$, then

${\displaystyle \lim _{n\to \infty }{\frac {\log(1+p_{n})}{p_{n}}}=\lim _{x\to 0}{\frac {\log(1+x)}{x}}=1,}$

and by the limit comparison test it follows that the two series

${\displaystyle \sum _{n=1}^{\infty }\log(1+p_{n})\quad {\text{and}}\quad \sum _{n=1}^{\infty }p_{n},}$

are equivalent meaning that either they both converge or they both diverge.

If the series ${\textstyle \sum _{n=1}^{\infty }\log(a_{n})}$ diverges to ${\displaystyle -\infty }$, then the sequence of partial products of the a_n converges to zero. The infinite product is said to diverge to zero. ^[1]

For the case where the ${\displaystyle p_{n}}$ have arbitrary signs, the convergence of the sum ${\textstyle \sum _{n=1}^{\infty }p_{n}}$ does not guarantee the convergence of the product ${\textstyle \prod _{n=1}^{\infty }(1+p_{n})}$. For example, if ${\displaystyle p_{n}={\frac {(-1)^{n}}{\sqrt {n}}}}$, then ${\textstyle \sum _{n=1}^{\infty }p_{n}}$ converges, but ${\textstyle \prod _{n=1}^{\infty }(1+p_{n})}$ diverges to zero. However, if ${\textstyle \sum _{n=1}^{\infty }|p_{n}|}$ is convergent, then the product ${\textstyle \prod _{n=1}^{\infty }(1+p_{n})}$ converges absolutely–that is, the factors may be rearranged in any order without altering either the convergence, or the limiting value, of the infinite product. ^[2] Also, if ${\textstyle \sum _{n=1}^{\infty }|p_{n}|^{2}}$ is convergent, then the sum ${\textstyle \sum _{n=1}^{\infty }p_{n}}$ and the product ${\textstyle \prod _{n=1}^{\infty }(1+p_{n})}$ are either both convergent, or both divergent. ^[3]

Product representations of functions

Main article: Weierstrass factorization theorem

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

${\displaystyle f(z)=z^{m}e^{\phi (z)}\prod _{n=1}^{\infty }\left(1-{\frac {z}{u_{n}}}\right)\exp \left\lbrace {\frac {z}{u_{n}}}+{\frac {1}{2}}\left({\frac {z}{u_{n}}}\right)^{2}+\cdots +{\frac {1}{\lambda _{n}}}\left({\frac {z}{u_{n}}}\right)^{\lambda _{n}}\right\rbrace }$

where λ_n are non-negative integers that can be chosen to make the product converge, and ${\displaystyle \phi (z)}$ is some entire 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 λ_n. However, for most functions, there will be some minimum non-negative integer p such that λ_n = 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

${\displaystyle f(z)=z^{m}e^{\phi (z)}\prod _{n=1}^{\infty }\left(1-{\frac {z}{u_{n}}}\right).}$

This can be regarded as a generalization of the fundamental theorem of algebra, since for polynomials, the product becomes finite and ${\displaystyle \phi (z)}$ is constant.

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

Function	Infinite product representation(s)	Notes
Simple pole	${\displaystyle {\begin{aligned}{\frac {1}{1-z}}&=\prod _{n=1}^{\infty }e^{z^{n}/n}\\&=\prod _{n=0}^{\infty }\left(1+z^{2^{n}}\right)\end{aligned}}}$
Sinc function	${\displaystyle {\frac {\sin \pi z}{\pi z}}=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)}$	This is due to Euler. Wallis' formula for π is a special case of this.
Reciprocal gamma function	${\displaystyle {\begin{aligned}{\frac {1}{\Gamma (z)}}&=ze^{\gamma z}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)e^{-{\frac {z}{n}}}\\&=z\prod _{n=1}^{\infty }{\frac {1+{\frac {z}{n}}}{\left(1+{\frac {1}{n}}\right)^{z}}}\end{aligned}}}$	Schlömilch [ clarification needed ]
Weierstrass sigma function	${\displaystyle \sigma (z)=z\prod _{\omega \in \Lambda _{*}}\left(1-{\frac {z}{\omega }}\right)e^{{\frac {z^{2}}{2\omega ^{2}}}+{\frac {z}{\omega }}}}$	Here ${\displaystyle \Lambda _{*}}$ is the lattice without the origin.
Q-Pochhammer symbol	${\displaystyle (z;q)_{\infty }=\prod _{n=0}^{\infty }(1-zq^{n})}$	Widely used in q-analog theory. The Euler function is a special case.
Ramanujan theta function	${\displaystyle {\begin{aligned}f(a,b)&=\sum _{n=-\infty }^{\infty }a^{\frac {n(n+1)}{2}}b^{\frac {n(n-1)}{2}}\\&=\prod _{n=0}^{\infty }(1+a^{n+1}b^{n})(1+a^{n}b^{n+1})(1-a^{n+1}b^{n+1})\end{aligned}}}$	An expression of the Jacobi triple product, also used in the expression of the Jacobi theta function
Riemann zeta function	${\displaystyle \zeta (z)=\prod _{n=1}^{\infty }{\frac {1}{1-p_{n}^{-z}}}}$	Here pn denotes the nth prime number. This is a special case of the Euler product.

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 denoted ζ(z)) on the whole complex plane except at the point z = 1, where it has a simple pole.

See also

• Infinite products in trigonometry
• Iterated binary operation
• Infinite expression
• Infinite series
• Pentagonal number theorem

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.
2. Trench, William F. (1999). "Conditional Convergence of Infinite Products" (PDF). American Mathematical Monthly. 106 (7): 646–651. doi:10.1080/00029890.1999.12005098 . Retrieved December 10, 2018.
3. Knopp, Konrad (1954). Theory and Application of Infinite Series. London: Blackie & Son Ltd.

• Knopp, Konrad (1990). Theory and Application of Infinite Series. Dover Publications. ISBN   978-0-486-66165-0.
• Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). Boston: McGraw Hill. ISBN   0-07-054234-1.
• Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Dover Publications. ISBN   978-0-486-61272-0.

External links

• Infinite products from Wolfram Math World
• A Collection of Infinite Products – I
• A Collection of Infinite Products – II