Infinite product
Encyclopedia
In mathematics
, for a sequence
of complex numbers a1, a2, a3, ... the infinite product
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 and John Wallis (Wallis product):
converges if and only if the sum
converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products.
For products in which each
, written as, for instance, 
,
where
, the bounds
show that the infinite product converges precisely if the infinite sum of the pn converges. This relies on the Monotone convergence theorem.
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
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
This can be regarded as a generalization of the Fundamental Theorem of Algebra
, since the product becomes finite and
is constant for polynomials.
In addition to these examples, the following representations are of special note:
Note that the last of these is not a product representation of the same sort discussed above, as ζ is not entire.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, for a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of complex numbers a1, a2, a3, ... the infinite product
is defined to be the limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
of the partial products a1a2...an as n increases without bound. The product is said to converge
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
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
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
, such as the following two products, respectively by Viète and John Wallis (Wallis product):
Convergence criteria
The product of positive real numbers with a finite amount ofconverges if and only if the sum
converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products.
For products in which each
, written as, for instance, ,where
, the boundsshow that the infinite product converges precisely if the infinite sum of the pn converges. This relies on the Monotone convergence theorem.
Product representations of functions
One important result concerning infinite products is that every entire functionEntire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
f(z) (that is, every function that is holomorphic
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
over the entire complex plane
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
) 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
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
This can be regarded as a generalization of the Fundamental Theorem of Algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
, since the product becomes finite and
is constant for polynomials.In addition to these examples, the following representations are of special note:
Sine Sine In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions.... function |  | Euler - Wallis' formula for π is a special case of this. |
Gamma function Gamma function In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers... |  | Schlömilch |
Weierstrass sigma function Weierstrass sigma function In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass.-Weierstrass sigma-function:... |  | Here  is the lattice without the origin. |
Q-Pochhammer symbol |  | Widely used in q-analog Q-analog Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1... theory. The Euler function is a special case. |
Ramanujan theta function Ramanujan theta function In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta... |   | An expression of the Jacobi triple product, also used in the expression of the Jacobi theta function |
Riemann zeta function |  | Here pn denotes the sequence of prime number Prime number A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2... s. This is a special case of the Euler product Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:... . |
Note that the last of these is not a product representation of the same sort discussed above, as ζ is not entire.
See also
- Infinite products in trigonometry
- Infinite seriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
- Continued fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
- Infinite expressionInfinite expression (mathematics)In mathematics, an infinite expression is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth...
- Iterated binary operationIterated binary operationIn mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the...