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Branch (mathematical Analysis)
In mathematics, a principal branch is a function which selects one branch point, branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are used in the definition of many inverse trigonometric functions, such as the selection either to define that :\arcsin:[-1,+1]\rightarrow\left[-\frac,\frac\right] or that :\arccos:[-1,+1]\rightarrow[0,\pi]. Exponentiation to fractional powers A more familiar principal branch function, limited to real numbers, is that of a positive real number raised to the power of . For example, take the relation , where is any positive real number. This relation can be satisfied by any value of equal to a square root of (either positive or negative). By convention, is used to denote the positive square root of . In this instance, the positive square root function is taken as the principal branch of the multi-valued relation . Complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include Graph of a function, graphs of Multivalued function, multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a Surface (topology), surface: a two-dimensional real manifold, but it contains more structure (specifically a Complex Manifold, complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and Metrizabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to be any complex number w for which e^w = z.Ahlfors, Section 3.4.Sarason, Section IV.9. Such a number w is denoted by \log z. If z is given in polar form as z = re^, where r and \theta are real numbers with r>0, then \ln r + i \theta is one logarithm of z, and all the complex logarithms of z are exactly the numbers of the form \ln r + i\left(\theta + 2\pi k\right) for integers k. These logarithms are equally spaced along a vertical line in the complex plane. * A complex-valued function \log \colon U \to \mathbb, defined on some subset U of the set \mathbb^* of nonzero complex numbers, satisfying e^ = z for all z in U. Such complex logarithm functions are analogous to the real logarithm function \ln \colon \mathbb_ \to \mathbb, which is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch Cut
In the mathematical field of complex analysis, a branch point of a multivalued function is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation w^2=z for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch (mathematical analysis), branch of that Function (mathematics), function, so that it is Single-valued function, single-valued. A simple case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as \sqrt. Motivation Consider the complex logarithm function . It is defined as the complex number such that :e^w = z. Now, for example, say we wish to find . This means we want to solve :e^w = i for w. The value i\pi/2 is a solution. However, there are other solutions, which is evidenced by considering the position of in the complex plane and in particular its argument (complex analysis), argument \arg i. We can rotate counterclockwise \pi/2 radians from 1 to reach initially, but if we rotate further another 2\pi we rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atan2
In computing and mathematics, the function (mathematics), function atan2 is the 2-Argument of a function, argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi 0, \\[5mu] \arctan\left(\frac y x\right) + \pi &\text x < 0 \text y \ge 0, \\[5mu] \arctan\left(\frac y x\right) - \pi &\text x < 0 \text y < 0, \\[5mu] +\frac &\text x = 0 \text y > 0, \\[5mu] -\frac &\text x = 0 \text y < 0, \\[5mu] \text &\text x = 0 \text y = 0. \end Instead of the tangent, it can be convenient to use the half-tangent as a representation of an angle, partly because the angle has a unique half-tangent, (See tangent half-angle formula.) The expression with in the denominator should be used when and to avoid possible loss of significance in computing . When an function is unavailable, it can be computed as twice the arctangent of the half-tangent . That is, |
Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , then is the logarithm of to base , written , so . As a single-variable function, the logarithm to base is the inverse of exponentiation with base . The logarithm base is called the ''decimal'' or ''common'' logarithm and is commonly used in science and engineering. The ''natural'' logarithm has the number as its base; its use is widespread in mathematics and physics because of its very simple derivative. The ''binary'' logarithm uses base and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written . Logarithms were introduced by John Napier in 1614 as a means of simplifying calculation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch Point
In the mathematical field of complex analysis, a branch point of a multivalued function is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation w^2=z for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4^2 = (-4)^2 = 16. Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'' or simply ''the square root'' (with a definite article, see below), which is denoted by \sqrt, where the symbol "\sqrt" is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative , the principal square root can also be written in exponent notation, as x^. Every positive number has two square roots: \sqrt (which is positive) and -\sqrt (which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
