|
Hölder's Theorem
In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found. The theorem also generalizes to the q -gamma function. Statement of the theorem For every n \in \N_0, there is no non-zero polynomial P \in \Complex ;Y_0,Y_1,\ldots,Y_n such that \forall z \in \Complex \setminus \Z _: \qquad P \left( z;\Gamma(z),\Gamma'(z),\ldots,(z) \right) = 0, where \Gamma is the gamma function. For example, define P \in \Complex ;Y_0,Y_1,Y_2 by P ~ \stackrel ~ X^2 Y_2 + X Y_1 + (X^2 - \nu^2) Y_0. Then the equation P \left (z;f(z),f'(z),f''(z) \right ) = z^2 f''(z) + z f'(z) + \left (z^2 - \nu^2 \right ) f(z) \equiv 0 is called an ''algebraic differential equation'', which, in this case, has the solutions f = J_ and f = Y_ — the Bessel functions of the first and second kind respec ... [...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] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Differential Equation
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to include equations formed by means of differential operators, in which the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra and number theory. A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator. Formulations *Derivations ''D'' can be used as algebraic analogues of the formal part of differential calculus, so that algebraic differential equations make sense in commutative rings. *The theory of differential fields was set u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Hölder
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart. Early life and education Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christian Gottlieb Hölder (1776–1847); his two brothers also became professors. He first studied at the ''Polytechnikum'' (which today is the University of Stuttgart) and then in 1877 went to Berlin where he was a student of Leopold Kronecker, Karl Weierstrass, and Ernst Kummer. In 1877, he entered the University of Berlin and took his doctorate from the University of Tübingen in 1882. The title of his doctoral thesis was "Beiträge zur Potentialtheorie" ("Contributions to potential theory"). Following this, he went to the University of Leipzig but was unable to habilitate there, instead earning a second doctorate and habilitation at the University of Göttingen, both in 1884. Academic career and later life He was unable to get government ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, Nigel Hitchin, and Thomas Schick. Currently, the managing editor of Mathematische Annalen is Yoshikazu Giga (University of Tokyo). Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947, the journal briefly ceased publication. References External links''Mathematische Annalen''homepage a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-gamma Function
In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by \Gamma_q(x) = (1-q)^\prod_^\infty \frac=(1-q)^\,\frac when , q, 1. Here (\cdot;\cdot)_\infty is the infinite q-Pochhammer symbol. The q-gamma function satisfies the functional equation \Gamma_q(x+1) = \frac\Gamma_q(x)= q\Gamma_q(x) In addition, the q-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (). For non-negative integers n, \Gamma_q(n)= -1q! where cdotq is the q-factorial function. Thus the q-gamma function can be considered as an extension of the q-factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit \lim_ \Gamma_q(x) = \Gamma(x). There is a simple proof of this limit by Gosper. See the appendix of (). Transformation properties The q-gamma function satis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendentally Transcendental Function
A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in \mathbb (the integers) and with algebraic initial conditions. History The term 'transcendentally transcendental' was introduced by E. H. Moore in 1896; the term 'hypertranscendental' was introduced by D. D. Morduhai-Boltovskoi in 1914. Definition One standard definition (there are slight variants) defines solutions of differential equations of the form :F\left(x, y, y', \cdots, y^ \right) = 0, where F is a polynomial with constant coefficients, as ''algebraically transcendental'' or ''differentially algebraic''. Transcendental functions which are not ''algebraically transcendental'' are ''transcendentally transcendental''. Hölder's theorem shows that the gamma function is in this category.Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", ''The American Mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Algebra
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one complex Zero of a function, root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field (mathematics), field of complex numbers is Algebraically closed field, algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, Degree of a polynomial, degree ''n'' polynomial with complex coefficients has, counted with Multiplicity (mathematics)#Multiplicity of a root of a polynomial, multiplicity, exactly ''n'' complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, it is not fundamental for modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Hugo Moll
Victor Hugo Moll (born 1956) is a Chilean American mathematician specializing in Analysis. Moll studied at the Universidad Santa Maria and at the New York University with a master's degree in 1982 and a doctorate in 1984 with Henry P. McKean (''Stability in the Large for Solitary Wave Solutions to McKean's Nerve Conduction Caricature''). He was a post-doctoral student at Temple University and became an assistant professor in 1986 and an associate professor in 1992 and in 2001 Professor at Tulane University. In 1990–1991, he was a visiting professor at the University of Utah, in 1999 at the Universidad Técnica Federico Santa María in Valparaíso, and in 1995 a visiting scientist at the Courant Institute of Mathematical Sciences of New York University. He deals with classical analysis, symbolic arithmetic and experimental mathematics, special functions and number theory. Projects Inspired by a 1988 paper in which proved several integrals in ''Table of Integrals, Series, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma And Related Functions
Gamma (; uppercase , lowercase ; ) 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 normally represents a voiced velar fricative , except before either of the two front vowels (/e/, /i/), where it represents 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 l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
