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Half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. Affine geometry The affine transformations of the upper half-plane include # shifts (x,y)\mapsto (x+c,y), c\in\mathbb, and # dilations (x,y)\mapsto (\lambda x,\lambda y), \lambda > 0. Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B. :Proof: First shift the center of to Then take \lambda=(\text\ B)/(\text\ A) and dilate. Then shift to the center of Inversive geometry Definition: \mathcal := \left\ . can be recognized as the circle of radius centered at and as the polar plot of \rho(\theta) = \cos \theta. Proposition: in and are collinear points. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Half-plane Model
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with coordinates whose coordinate is greater than zero, the upper half-plane, and a metric tensor (definition of distance) called the Poincaré metric is adopted, in which the local scale is inversely proportional to the coordinate. Points on the -axis, whose coordinate is equal to zero, represent ideal points (points at infinity), which are outside the hyperbolic plane proper. Sometimes the points of the half-plane model are considered to lie in the complex plane with positive imaginary part. Using this interpretation, each point in the hyperbolic plane is associated with a complex number. The half-plane model can be thought of as a map projection from the curved hyperbolic plane to the flat Euclidean plane. From the hyperboloid model (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-space (geometry)
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space is called a ''half-plane'' (open or closed). A half-space in a one-dimensional space is called a ''half-line'' or ray''.'' More generally, a half-space is either of the two parts into which a hyperplane divides an n-dimensional space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane. A half-space can be either ''open'' or ''closed''. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) ''upper half-space'' is the half-space of all (''x''1, ''x''2, ..., ''x''''n'') suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrant (plane Geometry)
The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. The axes themselves are, in general, not part of the respective quadrants. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the (''x''; ''y'') coordinates are I (+; +), II (−; +), III (−; −), and IV (+; −). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("northeast") quadrant. Mnemonic In the above graphic, the words in quotation marks are a mnemonic A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember. It makes use of e ... for remembering which three trigonometric functions (sine, cosine, tangent and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. Affine geometry The affine transformations of the upper half-plane include # shifts (x,y)\mapsto (x+c,y), c\in\mathbb, and # dilations (x,y)\mapsto (\lambda x,\lambda y), \lambda > 0. Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B. :Proof: First shift the center of to Then take \lambda=(\text\ B)/(\text\ A) and dilate. Then shift to the center of Inversive geometry Definition: \mathcal := \left\ . can be recognized as the circle of radius centered at and as the polar plot of \rho(\theta) = \cos \theta. Proposition: in and are collinear points. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Metric
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for ''q''-analogues are sometimes expressed. These various forms are reviewed below. Overview of metrics on Riemann surfaces A metric on the complex plane may be generally expressed in the form :ds^2=\lambda^2(z,\overline)\, dz\,d\overline where λ is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Mapping
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,Oxford English Dictionary, Draft Revision, Ju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Unit Disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose distance from ''P'' is less than or equal to one: :\bar D_1(P)=\.\, Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term ''unit disk'' is used for the open unit disk about the origin, D_1(0), with respect to the standard Euclidean metric. It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted \mathbb. The open unit disk, the plane, and the upper half-plane The function :f(z)=\frac is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of A Function
In mathematics, the domain of a function is the Set (mathematics), set of inputs accepted by the Function (mathematics), function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be". More precisely, given a function f\colon X\to Y, the domain of is . In modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both sets of real numbers, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the ''codomain'': the set to which all outputs must belong. The set of specific outputs the function assigns to elements of is called its ''Range of a function, range'' or ''Image (mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
