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Infinite Product
In mathematics, for a sequence 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 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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Factorization Theorem
In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem 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 the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. The theorem, which is named for Karl Weierstrass, is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. A generalization of the theorem extends it to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function. Motivation The consequences of the fundamental theorem of algebra are twofold.. Firstly, any finite sequence \ in the complex pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > 1 and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article " On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions". Throughout this article, (e^)^ sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Triple Product
In mathematics, the Jacobi triple product is the mathematical identity: :\prod_^\infty \left( 1 - x^\right) \left( 1 + x^ y^2\right) \left( 1 +\frac\right) = \sum_^\infty x^ y^, for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y'' ≠ 0. It was introduced by in his work ''Fundamenta Nova Theoriae Functionum Ellipticarum''. The Jacobi triple product identity is the Macdonald identity for the affine root system of type ''A''1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. Properties The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity. Let x=q\sqrt q and y^2=-\sqrt. Then we have :\phi(q) = \prod_^\infty \left(1-q^m \right) = \sum_^\infty (-1)^n q^. The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows: Let x=e^ and y=e^. Then the Jacobi theta function : \vartheta( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Theta Function
In mathematics, particularly -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. The function is named after mathematician Srinivasa Ramanujan. Definition The Ramanujan theta function is defined as :f(a,b) = \sum_^\infty a^\frac \; b^\frac for . The Jacobi triple product identity then takes the form :f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty. Here, the expression (a;q)_n denotes the -Pochhammer symbol. Identities that follow from this include :\varphi(q) = f(q,q) = \sum_^\infty q^ = and :\psi(q) = f\left(q,q^3\right) = \sum_^\infty q^\frac = and :f(-q) = f\left(-q,-q^2\right) = \sum_^\infty (-1)^n q^\frac = (q;q)_\infty This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Function
In mathematics, the Euler function is given by :\phi(q)=\prod_^\infty (1-q^k),\quad , q, A000203 On account of the identity \sum_ d = \sum_ \frac, this may also be written as :\ln(\phi(q)) = -\sum_^\infty \frac \sum_ d. Also if a,b\in\mathbb^+ and ab=\pi ^2, then :a^e^\phi (e^)=b^e^\phi (e^). Special values The next identities come from Ramanujan's Notebooks: : \phi(e^)=\frac : \phi(e^)=\frac : \phi(e^)=\frac : \phi(e^)=\frac(\sqrt-1)^ Using the Pentagonal number theorem, exchanging sum and integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ..., and then invoking complex-analytic methods, one derives : \int_0^1\phi(q)\,\mathrmq = \frac. References Notes Other * {{Leonhard Euler Number theory Q-analogs Leonhard Euler km:អនុគមន៍អឺលែ� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analog
In mathematics, 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 . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a prime power). ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Pochhammer Symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The relation between the sigma, zeta, and \wp functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant. Weierstrass sigma function The Weierstrass sigma function associated to a two-dimensional lattice \Lambda\subset\Complex is defined to be the product : \begin \operatorname&=z\prod_ \left(1-\frac\right) e^ \\ &=z\prod_^\infty \left(1-\frac\right) e^ \end where \Lambda^ denotes \Lambda-\ or \ are a ''fundamental pair of periods''. Through careful manipulation of the Weierstrass factorization theorem as it relates also to the sine function, another potentially more manageable infinite product definition is : \operatorname=\frace^\sin\prod_^\infty\left(1-\frac\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Gamma Function
In mathematics, the reciprocal gamma function is the function :f(z) = \frac, where denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that grows no faster than ), but of infinite type (meaning that grows faster than any multiple of , since its growth is approximately proportional to in the left-hand plane). The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function. Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem. Infinite product expansion Following from the infinite product definitions for the gamma function, due to Euler and Weierstrass respectively, we get the following infinite product expansion for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
