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Weierstrass Functions
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) \exp\left(\frac zw + \frac12\left(\frac zw\right)^2\right) \\ mu&= z\prod_^\infty \left(1 - \frac\right) \exp \end where \Lambda^ denotes \Lambda-\ and (\omega_1,\omega_2) is 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Logarithmic Derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the infinitesimal absolute change in , namely scaled by the current value of . When is a function of a real variable , and takes real, strictly positive values, this is equal to the derivative of , or the natural logarithm of . This follows directly from the chain rule: \frac\ln f(x) = \frac \frac Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Weierstrass Elliptic Function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy Cursive, script ''p''. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are Doubly_periodic_function, doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Symbol for Weierstrass \wp-function Motivation A Cubic_form, cubic of the form C_^\mathbb=\ , where g_2,g_3\in\mathbb are complex numbers with g_2^3-27g_3^2\neq0, cannot be Rational_variety, rationally parameterized. Yet one still wants to find a way to parameterize it. For the quadric K=\left\; the unit circle, there exists a (non-rational) parameterizatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dirichlet Eta Function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdots. This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ''ζ''(''s'') — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ''ζ''*(''s''). The following relation holds: \eta(s) = \left(1-2^\right) \zeta(s) Both the Dirichlet eta function and the Riemann zeta function are special cases of polylogarithms. While the Dirichlet series expansion for the eta function is convergent only for any complex number ''s'' with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is entire and \eta( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dedekind Eta Function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory. Definition For any complex number with , let ; then the eta function is defined by, :\eta(\tau) = e^\frac \prod_^\infty \left(1-e^\right) = q^\frac \prod_^\infty \left(1 - q^n\right) . Raising the eta equation to the 24th power and multiplying by gives :\Delta(\tau)=(2\pi)^\eta^(\tau) where is the modular discriminant. The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice. The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it. The eta function satisfies the functional equations :\begin \eta(\tau+1) &=e^\frac\eta(\tau),\\ \eta\left(-\frac\right) &= \sqrt\, \eta(\tau).\, \end In the second equation the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 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 many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Eisenstein Series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms. Eisenstein series for the modular group Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the following series: :G_(\tau) = \sum_ \frac. This series absolutely converges to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its -covariance. Explicitly if and then :G_ \left( \frac \right) = (c\tau +d)^ G_(\tau) Note that is necessary such that the series converges absolutely, whereas needs to be even otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Weierstrass Zeta 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) \exp\left(\frac zw + \frac12\left(\frac zw\right)^2\right) \\ mu&= z\prod_^\infty \left(1 - \frac\right) \exp \end where \Lambda^ denotes \Lambda-\ and (\omega_1,\omega_2) is 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 inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Special Function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 Zero of a function, 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 It is clear that any finite set \ of points in the complex plane has an associated polynomial p(z) = \pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fundamental Pair Of Periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Definition A fundamental pair of periods is a pair of complex numbers \omega_1,\omega_2 \in \Complex such that their ratio \omega_2 / \omega_1 is not real. If considered as vectors in \R^2, the two are linearly independent. The lattice generated by \omega_1 and \omega_2 is :\Lambda = \left\. This lattice is also sometimes denoted as \Lambda(\omega_1, \omega_2) to make clear that it depends on \omega_1 and \omega_2. It is also sometimes denoted by \Omega\vphantom or \Omega(\omega_1, \omega_2), or simply by (\omega_1, \omega_2). The two generators \omega_1 and \omega_2 are called the ''lattice basis''. The parallelogram with vertices (0, \omega_1, \omega_1+\omega_2, \omega_2) is called the ''fundamental parallelogram''. While a fundamental pair gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
