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Doubly Periodic Functions
In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers ''u'' and ''v'' that are linearly independent as vectors over the field of real numbers. That ''u'' and ''v'' are periods of a function ''ƒ'' means that :f(z + u) = f(z + v) = f(z)\, for all values of the complex number ''z''. The doubly periodic function is thus a two-dimensional extension of the simpler singly periodic function, which repeats itself in a single dimension. Familiar examples of functions with a single period on the real number line include the trigonometric functions like cosine and sine, In the complex plane the exponential function ''e''''z'' is a singly periodic function, with period 2''πi''. Examples As an arbitrary mapping from pairs of reals (or complex numbers) to reals, a doubly periodic function can be constructed with little effort. For example, assume that the periods are 1 and ''i'', so that the repe ... [...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] |
Pole (complex Analysis)
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if it is a zero of the function and is holomorphic in some neighbourhood of (that is, complex differentiable in a neighbourhood of ). A function is meromorphic in an open set if for every point of there is a neighborhood of in which either or is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros. Definitions A function of a complex variable is holomorphic in an open domai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Pair Of Periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define 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 ω2/ω1 is not real. If considered as vectors in \mathbb^2, the two are not collinear. The lattice generated by ω1 and ω2 is :\Lambda = \left\ This lattice is also sometimes denoted as Λ(''ω''1, ''ω''2) to make clear that it depends on ω1 and ω2. It is also sometimes denoted by Ω or Ω(''ω''1, ''ω''2), or simply by ⟨''ω''1, ''ω''2⟩. The two generators ω1 and ω2 are called the ''lattice basis''. The parallelogram defined by the vertices 0, \omega_1 and \omega_2 is called the ''fundamental parallelogram''. While a fundamental pair generates a lattice, a lattice does not have any unique fundamental pair; in fact, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dixon Elliptic Functions
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these functions satisfy the identity \operatorname^3 z + \operatorname^3 z = 1, as real functions they parametrize the cubic Fermat curve x^3 + y^3 = 1, just as the trigonometric functions sine and cosine parametrize the unit circle x^2 + y^2 = 1. They were named sm and cm by Alfred Dixon in 1890, by analogy to the trigonometric functions sine and cosine and the Jacobi elliptic functions sn and cn; Göran Dillner described them earlier in 1873. Definition The functions sm and cm can be defined as the solutions to the initial value problem: :\frac \operatorname z = -\operatorname^2 z,\ \frac \operatorname z = \operatorname^2 z,\ \operatorname(0) = 1,\ \operatorname(0) = 0 Or as the inverse of the Schwarz–Christoffel mapping from the complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate Elliptic Functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols and (sometimes the symbols and or and are used instead) are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle x^2+y^2 = x, the lemniscate sine relates the arc length to the chord length of a lemniscate \bigl(x^2+y^2\bigr)^2=x^2-y^2. The lemniscate functions have periods related to a number called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the ( quadratic) , ratio of perimeter to diameter of a circle. As complex functions, and have a square period lattice (a mul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Elliptic Functions
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 are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script ''p''. They play an important role in the theory of elliptic functions. 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 Definition Let \omega_1,\omega_2\in\mathbb be two complex numbers that are linearly independent over \mathbb and let \Lambda:=\mathbb\omega_1+\mathbb\omega_2:=\ be the lattice generated by those numbers. Then the \wp-function is defined as follows: \weierp(z,\omega_1,\omega_2):=\weierp(z,\Lambda) := \frac + \sum_\left(\frac 1 - \frac 1 \right). This series converges locally uniformly absolutely in \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Elliptic Functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometry, trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Elliptic Functions
In mathematics Abel elliptic functions are a special kind of elliptic functions, that were established by the Norwegian mathematician Niels Henrik Abel. He published his paper "Recherches sur les Fonctions elliptiques" in Crelle's Journal in 1827. It was the first work on elliptic functions that was actually published. Abel's work on elliptic functions also influenced Jacobi's studies of elliptic functions, whose 1829 published book " Fundamenta nova theoriae functionum ellipticarum" became the standard work on elliptic functions. History Abels starting point were the elliptic integrals which had been studied in great detail by Adrien-Marie Legendre. He began his research in 1823 when he still was a student. In particular he viewed them as complex functions which at that time were still in their infancy. In the following years Abel continued to explore these functions. He also tried to generalize them to functions with even more periods, but seemed to be in no hurry to publish h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass \wp-function. Further development of this theory led to hyperelliptic functions and modular forms. Definition A meromorphic function is called an elliptic function, if there are two \mathbb- linear independent complex numbers \omega_1,\omega_2\in\mathbb such that : f(z + \omega_1) = f(z) and f(z + \omega_2) = f(z), \quad \forall z\in\mathbb. So elliptic functions have two periods and are therefore also called ''doubly periodic''. Period lattice and fundamental domain Iff is an elliptic function with periods \omega_1,\omega_2 it also holds that : f(z+\gamma)=f(z) for every linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeros And Poles
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if it is a zero of the function and is holomorphic in some neighbourhood of (that is, complex differentiable in a neighbourhood of ). A function is meromorphic in an open set if for every point of there is a neighborhood of in which either or is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros. Definitions A function of a complex variable is holomorphic in an open domai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem. Statement The statement is as follows: Let be a simply connected open subset of the complex plane containing a finite list of points , , and a function defined and holomorphic on . Let be a closed rectifiable curve in , and denote the winding number of around by . The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point: \oint_\gamma f(z)\, dz = 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ). If is a positively oriented simple closed curve, if i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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