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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, for a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of the partial products ''a''1''a''2...''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 John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
( Wallis product): :\frac = \frac \cdot \frac \cdot \frac \cdot \; \cdots :\frac = \left(\frac \cdot \frac\right) \cdot \left(\frac \cdot \frac\right) \cdot \left(\frac \cdot \frac\right) \cdot \left(\frac \cdot \frac\right) \cdot \; \cdots = \prod_^ \left( \frac \right).


Convergence criteria

The product of positive real numbers :\prod_^ a_n converges to a nonzero real number if and only if the sum :\sum_^ \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 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_^ p_n \le \prod_^ \left( 1 + p_n \right) \le \exp \left( \sum_^p_n \right) show that the infinite product 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 p_n \to 0, then :\lim_ \frac = \lim_ \frac = 1, and by the limit comparison test it follows that the two series :\sum_^\infty \log(1+p_n) \quad \text \quad \sum_^\infty p_n, are equivalent meaning that either they both converge or they both diverge. The same proof also shows that if a_n = 1 - q_n for some 0\le q_n < 1, then \prod_^\infty (1-q_n) converges to a non-zero number if and only if \sum_^\infty q_n converges. If the series \sum_^ \log(a_n) diverges to -\infty, then the sequence of partial products of the ''a''''n'' converges to zero. The infinite product is said to diverge to zero. For the case where the p_n have arbitrary signs, the convergence of the sum \sum_^\infty p_n does not guarantee the convergence of the product \prod_^\infty (1+p_n). For example, if p_n = \frac, then \sum_^\infty p_n converges, but \prod_^\infty (1 + p_n) diverges to zero. However, if \sum_^\infty , p_n, is convergent, then the product \prod_^\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. Also, if \sum_^\infty , p_n, ^2 is convergent, then the sum \sum_^\infty p_n and the product \prod_^\infty (1+p_n) are either both convergent, or both divergent.


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 In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
) 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 ''u''1, ''u''2, ''u''3, ... (listed with multiplicities equal to their orders), then :f(z) = z^m e^ \prod_^ \left(1 - \frac \right) \exp \left\lbrace \frac + \frac\left(\frac\right)^2 + \cdots + \frac \left(\frac\right)^ \right\rbrace where ''λ''''n'' are non-negative integers that can be chosen to make the product converge, and \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 :f(z) = z^m e^ \prod_^ \left(1 - \frac\right). This can be regarded as a generalization of the fundamental theorem of algebra, since for polynomials, the product becomes finite and \phi (z) is constant. In addition to these examples, the following representations are of special note: 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 In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
, 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 * Continued fraction * Iterated binary operation * Infinite expression * Infinite series * Pentagonal number theorem


References

* * * {{Cite book , editor1-last=Abramowitz , editor1-first=Milton , editor1-link=Milton Abramowitz , editor2-last=Stegun , editor2-first=Irene A. , editor2-link=Irene Stegun , title= Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , publisher=
Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...
, year=1972 , isbn=978-0-486-61272-0


External links


Infinite products from Wolfram Math WorldA Collection of Infinite Products – IA Collection of Infinite Products – II
Sequences and series Mathematical analysis Multiplication es:Productorio