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Super-logarithm
In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverse functions, super-roots and super-logarithms. There are several ways of interpreting super-logarithms: * As the Abel function of exponential functions, * As the inverse function of tetration with respect to the height, * As a generalization of Robert Munafo'large number class system For positive integer values, the super-logarithm with base-'' e'' is equivalent to the number of times a logarithm must be iterated to get to 1 (the Iterated logarithm). However, this is not true for negative values and so cannot be considered a full definition. The precise definition of the super-logarithm depends on a precise definition of non-integral tetration (that is, for ''y'' not an integer). There is no clear consensus on the definition of non-integral tetration and so there is likewise no clear consensus o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as repeated exponentiation, means , where ' copies of ' are iterated via exponentiation, right-to-left, i.e. the application of exponentiation n-1 times. ' is called the "height" of the function, while ' is called the "base," analogous to exponentiation. It would be read as "the th tetration of ". It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is also defined recursively as : := \begin 1 &\textn=0, \\ a^ &\textn>0, \end allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers. The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super-root
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as repeated exponentiation, means , where ' copies of ' are iterated via exponentiation, right-to-left, i.e. the application of exponentiation n-1 times. ' is called the "height" of the function, while ' is called the "base," analogous to exponentiation. It would be read as "the th tetration of ". It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is also defined recursively as : := \begin 1 &\textn=0, \\ a^ &\textn>0, \end allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers. The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as repeated exponentiation, means , where ' copies of ' are iterated via exponentiation, right-to-left, i.e. the application of exponentiation n-1 times. ' is called the "height" of the function, while ' is called the "base," analogous to exponentiation. It would be read as "the th tetration of ". It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is also defined recursively as : := \begin 1 &\textn=0, \\ a^ &\textn>0, \end allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers. The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithms
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation. 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 frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (mathematical constant), as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base and is frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Logarithm
In computer science, the iterated logarithm of n, written n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition is the result of this recurrence relation: : \log^* n := \begin 0 & \mbox n \le 1; \\ 1 + \log^*(\log n) & \mbox n > 1 \end On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent: :\log^* n = \lceil \mathrm _e(n) \rceil i.e. the base ''b'' iterated logarithm is \log^* n = y if n lies within the interval ^b on the ''x''-axis. In computer science, is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base 2) instead of the natural logarithm (with base ''e''). Mathematically, the iterated logarithm is well-defined for any base greater than e^ \approx 1.444667, not only for base 2 and base ''e''. Analysis o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Approximation
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch Point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English language, English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer (Heidelberg University, Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics) (it Is A -1 And 0 Branch Function W-Lambert In Point -W(-1) )
Fixed point may refer to: * Fixed point (mathematics), a value that does not change under a given transformation * Fixed-point arithmetic, a manner of doing arithmetic on computers * Fixed point, a benchmark (surveying) The term benchmark, bench mark, or survey benchmark originates from the chiseled horizontal marks that surveyors made in stone structures, into which an angle-iron could be placed to form a "bench" for a leveling rod, thus ensuring that a le ... used by geodesists * Fixed point join, also called a recursive join * Fixed point, in quantum field theory, a coupling where the beta function vanishes – see {{disambiguation, math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Extension
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
