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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] |
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] |
Global Analytic Function
In the mathematical field of complex analysis, a global analytic function is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analytic functions arise naturally in considering the possible analytic continuations of an analytic function, since analytic continuations may have a non-trivial monodromy. They are one foundation for the theory of Riemann surfaces. Definition The following definition is in , but also found in Weyl or perhaps Weierstrass. An analytic function in an open set ''U'' is called a function element. Two function elements (''f''1, ''U''1) and (''f''2, ''U''2) are said to be analytic continuations of one another if ''U''1 ∩ ''U''2 ≠ ∅ and ''f''1 = ''f''2 on this intersection. A chain of analytic continuations is a finite sequence of function elements (''f''1, ''U''1), …, (''f''''n'',''U''''n'') such that each consecutive pair are analytic continuations of one anot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface Log
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Early years Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Function
In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: * f(x) = 1/x * f(x) = \sqrt * f(x) = \frac Some algebraic functions, however, cannot be expressed by such finite expressions (this is the Abel–Ruffini theorem). This is the case, for example, for the Bring radical, which is the function implicitly defined by : f(x)^5+f(x)+x = 0. In more precise terms, an algebraic function of degree in one variable is a function y = f(x), that is continuous in its domain and satisfies a polynomial equation of positive degree : a_n(x)y^n+a_(x)y^+\cdots+a_0(x)=0 where the coefficients are polynomial functions of , with integer coefficients. It can be shown that the same class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is the line segment or distance from its center to any of its Vertex (geometry), vertices. The name comes from the Latin ''radius'', meaning ray but also the spoke of a chariot wheel.Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08. The typical abbreviation and mathematical symbol for radius is ''R'' or ''r''. By extension, the diameter ''D'' is defined as twice the radius:Definition of radius at mathwords.com. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...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] |
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] |
Annulus (mathematics)
In mathematics, an annulus (: annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' meaning 'little ring'. The adjectival form is ''annular'' (as in annular eclipse). The open annulus is topologically equivalent to both the open cylinder and the punctured plane. Area The area of an annulus is the difference in the areas of the larger circle of radius and the smaller one of radius : :A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right) = \pi (R+r)(R-r) . The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so and are sides of a right-angled triangle with hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
