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Bohr–Mollerup Theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for by :\Gamma(x)=\int_0^\infty t^ e^\,dt as the ''only'' positive function , with domain on the interval , that simultaneously has the following three properties: * , and * for and * is logarithmically convex. A treatment of this theorem is in Artin's book ''The Gamma Function'', which has been reprinted by the AMS in a collection of Artin's writings. The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved. Statement :Bohr–Mollerup Theorem. is the only function that satisfies with convex and also with . Proof Let be a function with the assumed properties established above: and is convex, and . From we can establish :\Gamma(x+n)=(x+n-1)(x+n-2)(x+n-3)\cdots(x+1)x\Gamma(x) The purpose of the st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harald Bohr
Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr. He was a member of the Danish national football team for the 1908 Summer Olympics, where he won a silver medal. Biography Bohr was born in 1887 to Christian Bohr, a professor of physiology, from a Lutheran background, and Ellen Adler Bohr, a woman from a wealthy Jewish family of local renown. Harald had a close relationship with his elder brother, which ''The Times'' likened to that between Captain Cuttle and Captain Bunsby in Charles Dickens' ''Dombey and Son''. Mathematical career Like his father and brother before him, in 1904 Bohr enrolled at the University of Copenhagen, where he studied mathematics, obtaining his master's degree in 1909 and his doctorate a year later. Among his tutors were Hie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Mollerup
Johannes Mollerup (3 December 1872 – 27 June 1937) was a Danish mathematician. Gyldendal – Den Store Danske. Mollerup studied at the , and received his doctorate in 1903. Together with , he developed the which provides an easy |
Characterization (mathematics)
In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X'' have property ''P'', but that ''X'' is the ''only'' thing that has property ''P'' (i.e., ''P'' is a defining property of ''X''). Similarly, a set of properties ''P'' is said to characterize ''X'', when these properties distinguish ''X'' from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of ''X'' in terms of ''P'' include "''P'' is necessary and sufficient for ''X''", and "''X'' holds if and only if ''P''". It is also common to find statements such as "Property ''Q'' characterizes ''Y'' up to isomorphism". The first type of statement says in different words that the extension of ''P'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Convexity
In mathematics, a function (mathematics), function ''f'' is logarithmically convex or superconvex if \circ f, the function composition, composition of the logarithm with ''f'', is itself a convex function. Definition Let be a convex set, convex subset of a real numbers, real vector space, and let be a function taking negative and positive numbers, non-negative values. Then is: * Logarithmically convex if \circ f is convex, and * Strictly logarithmically convex if \circ f is strictly convex. Here we interpret \log 0 as -\infty. Explicitly, is logarithmically convex if and only if, for all and all , the two following equivalent conditions hold: :\begin \log f(tx_1 + (1 - t)x_2) &\le t\log f(x_1) + (1 - t)\log f(x_2), \\ f(tx_1 + (1 - t)x_2) &\le f(x_1)^tf(x_2)^. \end Similarly, is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all . The above definition permits to be zero, but if is logarithmically convex and vanis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma had ... [...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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wielandt Theorem
In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers z for which \mathrm\,z > 0 by :\Gamma(z)=\int_0^ t^ \mathrm e^\,\mathrm dt, as the only function f defined on the half-plane H := \ such that: * f is holomorphic on H; * f(1)=1; * f(z+1)=z\,f(z) for all z \in H and * f is bounded on the strip \. This theorem named after the mathematician Helmut Wielandt. See also * Bohr–Mollerup theorem * Hadamard's gamma function References * {{cite journal, author=Reinhold Remmert, title=Wielandt's theorem about the {{math, Γ-function, journal=American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ..., volume=103, year=1996, pages=214–220, jstor=2975370. Gamma and related functions Theorems in complex analysis [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma And Related Functions
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter represents either a voiced velar fricative or a voiced palatal fricative (while /g/ in foreign words is instead commonly transcribed as γκ). In the International Phonetic Alphabet and other modern Latin-alphabet based phonetic notations, it represents the voiced velar fricative. History The Greek letter Gamma Γ is a grapheme derived from the Phoenician letter (''gīml'') which was rotated from the right-to-left script of Canaanite to accommodate the Greek language's writing system of left-to-right. The Canaanite grapheme represented the /g/ phoneme in the Canaanite language, and as such is cognate with ''gimel'' ג of the Hebrew alphabet. Based on its name, the letter has been interpreted as an abstract representation of a camel's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
