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Beta-function
The beta function, also called the Euler beta function or the Euler integral of the first kind, is a special function in mathematics. Other meanings Beta function may also refer to: *Beta function (physics), details the running of the coupling strengths *Dirichlet beta function, closely related to the Riemann zeta function *Gödel's β function, used in mathematical logic to encode sequences of natural numbers *Beta function (accelerator physics), related to the transverse beam size at a given point in a beam transport system See also *Beta distribution {{dab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \operatorname(z_1), \operatorname(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.. Specifically, see 6.2 Beta Function. A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function that : \Beta(m,n) =\fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function (physics)
In theoretical physics, specifically quantum field theory, a beta function or Gell-Mann–Low function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is defined by the Gell-Mann–Low equation or renormalization group equation, given by :: \beta(g) = \mu \frac = \frac ~, and, because of the underlying renormalization group, it has no explicit dependence on ''μ'', so it only depends on ''μ'' implicitly through ''g''. This dependence on the energy scale thus specified is known as the running of the coupling parameter, a fundamental feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques. The concept of beta function was first introduced by Ernst Stueckelberg and André Petermann in 1953, and independently postulated by Murray Gell-Mann and Francis E. Low in 1954. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Beta Function
In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four. Definition The Dirichlet beta function is defined as :\beta(s) = \sum_^\infty \frac , or, equivalently, :\beta(s) = \frac\int_0^\frac\,dx. In each case, it is assumed that Re(''s'') > 0. Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex ''s''-plane: :\beta(s) = 4^ \left( \zeta\left(s,\right)-\zeta\left( s, \right) \right). Another equivalent definition, in terms of the Lerch transcendent, is: :\beta(s) = 2^ \Phi\left(-1,s,\right), which is once again valid for all complex values of ''s''. The Dirichlet beta function can also be written in terms of the polylogarithm function: :\beta(s) = \frac \left(\text_s(-i)-\text_s(i)\right). Also the series r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's β Function
In mathematical logic, Gödel's ''β'' function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The ''β'' function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The ''β'' function was introduced without the name in the proof of the first of Gödel's incompleteness theorems (Gödel 1931). The ''β'' function lemma given below is an essential step of that proof. Gödel gave the ''β'' function its name in (Gödel 1934). Definition The \beta function takes three natural numbers as arguments. It is defined as :\beta(x_1, x_2, x_3) = \mathrm(x_1, 1 + (x_3 + 1) \cdot x_2) = \mathrm(x_1, (x_3 \cdot x_2 + x_2 + 1) ), where \mathrm(x, y) denotes the remainder after integer division of x by y (Mendelson 1997:186). Special schema without parameters The ''β'' function is ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function (accelerator Physics)
The beta function in accelerator physics is a function related to the transverse size of the particle beam at the location s along the nominal beam trajectory. It is related to the transverse beam size as follows: \sigma(s) = \sqrt where * s is the location along the nominal beam trajectory * the beam is assumed to have a Gaussian shape in the transverse direction * \sigma(s) is the width parameter of this Gaussian * \epsilon is the RMS geometrical beam emittance, which is normally constant along the trajectory when there is no acceleration Typically, separate beta functions are used for two perpendicular directions in the plane transverse to the beam direction (e.g. horizontal and vertical directions). The beta function is one of the Courant–Snyder parameters (also called Twiss parameters). Beta star The value of the beta function at an interaction point is referred to as beta star. The beta function is typically adjusted to have a local minimum at such points (in ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
