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Ramanujan's Master Theorem
In mathematics, Ramanujan's Master Theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform of an analytic function. The result is stated as follows: If a complex-valued function f(x) has an expansion of the form : f(x)=\sum_^\infty \frac(-x)^k then the Mellin transform of f(x) is given by : \int_0^\infty x^ f(x) \, dx = \Gamma(s)\,\varphi(-s) where \Gamma(s) is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams). A similar result was also obtained by Glaisher. Alternative formalism An alternative formulation of Ramanujan's Master Theorem is as follows: : \int_0^\infty x^\left(\,\lambda(0) - x\,\lambda(1) + x^2\,\lambda(2) -\,\cdots\,\right) dx = \frac\,\lambda(-s) which gets converted to the above form after substituting \lambda(n) \equiv \frac a ... [...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 ... [...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 consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Function
In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term ''affine function'' is often used. * In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map. As a polynomial function In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). When the function is of only one variable, it is of the form :f(x)=ax+b, where and are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. is frequently referred to as the slope of the line, and as the intercept. If ''a > 0'' then the gradient is positive and the gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Equation
In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients a_1, \ldots, a_n are required to not all be zero. Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken. The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. In the case of just one variable, there is exactly one solution (provided that a_1\ne 0). Often, the term ''linear equation'' refers implicitly to this particular case, in which the variable is sensibly called the ''unknown''. In the case of two var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Independence
In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. Definition A sequence of vectors \mathbf_1, \mathbf_2, \dots, \mathbf_k from a vector space is said to be ''linearly dependent'', if there exist scalars a_1, a_2, \dots, a_k, not all zero, such that :a_1\mathbf_1 + a_2\mathbf_2 + \cdots + a_k\mathbf_k = \mathbf, where \mathbf denotes the zero vector. This implies that at least one of the scalars is nonzero, say a_1\ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a specia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. ''Parameter'' has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. Modelization When a system is modeled by equations, the values that describe the system are called ''parameters''. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division). The resulting so-called ''series'' often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions. Types of series expansions There are several kinds of series expansions, listed below. A ''Taylor series'' is a power series based on a function's derivatives at a single point. More specifically, if a function f: U\to\mathbb is infinitely differentiable around ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Formula
In mathematics, a reflection formula or reflection relation for a function ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a functional equation, and it is very common in the literature to use the term "functional equation" when "reflection formula" is meant. Reflection formulas are useful for numerical computation of special functions. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the complex plane) can be employed for all arguments. Known formulae The even and odd functions satisfy by definition simple reflection relations around ''a'' = 0. For all even functions, :f(-x) = f(x), and for all odd functions, :f(-x) = -f(x). A famous relationship is Euler's reflection formula :\Gamma(z)\Gamma(1-z) = \frac, \qquad z \not\in \mathbb Z for the gamma function \Gamma(z), due to Leonhard Euler. There is also a reflect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duplication Formula
Duplication, duplicate, and duplicator may refer to: Biology and genetics * Gene duplication, a process which can result in free mutation * Chromosomal duplication, which can cause Bloom and Rett syndrome * Polyploidy, a phenomenon also known as ''ancient genome duplication'' * Enteric duplication cysts, certain portions of the gastrointestinal tract * Diprosopus, a form of cojoined twins also known as ''craniofacial duplication'' * Diphallia, a medical condition also known as ''penile duplication'' Computing * Duplicate code, a source code sequence that occurs more than once in a program * Duplicate characters in Unicode, pairs of single Unicode code points that are canonically equivalent. The reason for this are compatibility issues with legacy systems * Data redundancy, either wanted or unwanted (in which case one resorts to data deduplication) * Content copying through cut, copy, and paste * File copying Mathematics * Duplication matrix, a linear transformation dealing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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