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 (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product):
mathematics, for a sequence of complex numbers a1, a2, a3, ... the infiniteproduct ∏ n = 1 ∞ a n = a 1 a 2 a 3 ⋯ {\displaystyle \prod _{n=1}^{\infty...
{(-1)^{n}x^{2n}}{(2n)!}}.} For applications to special functions, the following infiniteproduct formulae for trigonometric functions are useful: sin x = x ∏ n =...
theory, an Euler product is an expansion of a Dirichlet series into an infiniteproduct indexed by prime numbers. The original such product was given for...
hold even when the left-hand product contains zeros or poles. By taking limits, certain rational products with infinitely many factors can be evaluated...
{8}{9}}{\Big )}\cdot \;\cdots \\\end{aligned}}} Wallis derived this infiniteproduct using interpolation, though his method is not regarded as rigorous...
from the infiniteproduct definitions for the gamma function, due to Euler and Weierstrass respectively, we get the following infiniteproduct expansion...
The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q = 0, yielding ln (...
input series a way of expressing an entire function of finite order an infiniteproduct expansion for the Riemann zeta function This disambiguation page lists...
asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of...
above by the infinity symbol ∞. The product of such an infinite sequence is defined as the limit of the product of the first n terms, as n grows without...
10^{-43}} Schmuland has given appealing probabilistic formulations of the infiniteproduct Borwein integrals. For example, consider the random harmonic series...
_{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.} The following infiniteproduct for the sine is of great importance in complex analysis: sin z =...
so on. The set A × B is infinite if either A or B is infinite, and the other set is not the empty set. The Cartesian product can be generalized to the...
_{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).} Alternatively, the infiniteproduct for the sine can be proved using complex Fourier series. sin(z) is...
infinite series to estimate π to 11 digits around 1400. In 1593, François Viète published what is now known as Viète's formula, an infiniteproduct (rather...
Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series)...
looking for: ∏ n = 1 ∞ {\displaystyle \prod _{n=1}^{\infty }} – the Infiniteproduct of a sequence Capital pi notation This disambiguation page lists articles...
{\displaystyle \textstyle \prod _{0<i<j<n}j-i} . 2. Denotes an infiniteproduct. For example, the Euler product formula for the Riemann zeta function is ζ ( z ) =...
direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The...
be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The...
Halo Infinite is a 2021 first-person shooter game developed by 343 Industries and published by Xbox Game Studios. It is the sixth mainline installment...
{3n^{2}-n}{2}}.} The Jacobi Triple Product also allows the Jacobi theta function to be written as an infiniteproduct as follows: Let x = e i π τ {\displaystyle...
1). The normalized sinc function has a simple representation as the infiniteproduct: sin ( π x ) π x = ∏ n = 1 ∞ ( 1 − x 2 n 2 ) {\displaystyle {\frac...
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