In
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
:
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):
:
:
Convergence criteria
The product of positive real numbers
:
converges to a nonzero real number if and only if the sum
:
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 ''a
n'' fall outside the domain of ln, whereas finitely many such ''a
n'' can be ignored in the sum.
For products of reals in which each
, written as, for instance,
, where
, the bounds
:
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
, then
:
and by the
limit comparison test it follows that the two series
:
are equivalent meaning that either they both converge or they both diverge.
The same proof also shows that if
for some
, then
converges to a non-zero number if and only if
converges.
If the series
diverges to
, 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
have arbitrary signs, the convergence of the sum
does not guarantee the convergence of the product
. For example, if
, then
converges, but
diverges to zero. However, if
is convergent, then the product
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
is convergent, then the sum
and the product
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
:
where ''λ''
''n'' are non-negative integers that can be chosen to make the product converge, and
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
:
This can be regarded as a generalization of the
fundamental theorem of algebra, since for polynomials, the product becomes finite and
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