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Euler-Mascheroni Constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by : \begin \gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\ px&=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx. \end Here, represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: History The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and for the constant. The Italian mathematician Lorenzo Mascheroni attempted to calculate the constant to 32 dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Number
The number is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number is of great importance in mathematics, alongside 0, 1, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients. To 30 decimal places, the value of is: Definitions T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Beta Function
In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four. Definition The Dirichlet beta function is defined as :\beta(s) = \sum_^\infty \frac , or, equivalently, :\beta(s) = \frac\int_0^\frac\,dx. In each case, it is assumed that Re(''s'') > 0. Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex ''s''-plane: :\beta(s) = 4^ \left( \zeta\left(s,\right)-\zeta\left( s, \right) \right). Another equivalent definition, in terms of the Lerch transcendent, is: :\beta(s) = 2^ \Phi\left(-1,s,\right), which is once again valid for all complex values of ''s''. The Dirichlet beta function can also be written in terms of the polylogarithm function: :\beta(s) = \frac \left(\text_s(-i)-\text_s(i)\right). Also the series r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particular Values Of The Riemann Zeta Function
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted \zeta(s) and is named after the mathematician Bernhard Riemann. When the argument s is a real number greater than one, the zeta function satisfies the equation \zeta(s) = \sum_^\infty\frac \, . It can therefore provide the sum of various convergent infinite series, such as \zeta(2) = \frac + \frac + \frac + \ldots \, . Explicit or numerically efficient formulae exist for \zeta(s) at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments. The same equation in s above also holds when s is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stieltjes Constants
In mathematics, the Stieltjes constants are the numbers \gamma_k that occur in the Laurent series expansion of the Riemann zeta function: :\zeta(1+s)=\frac+\sum_^\infty \frac \gamma_n s^n. The constant \gamma_0 = \gamma = 0.577\dots is known as the Euler–Mascheroni constant. Representations The Stieltjes constants are given by the limit : \gamma_n = \lim_ \left\ = \lim_ . (In the case ''n'' = 0, the first summand requires evaluation of 00, which is taken to be 1.) Cauchy's differentiation formula leads to the integral representation :\gamma_n = \frac \int_0^ e^ \zeta\left(e^+1\right) dx. Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors. In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that : \gamma_n = \frac\delta_+\frac\int_0^\infty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894. Definition The Laurent series for a complex function f(z) about an arbitrary point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where the coefficients a_n are defined by a contour integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic ( analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing and Concave function, strictly concave on (0,\infty), and it Asymptotic analysis, asymptotically behaves as :\psi(z) \sim \ln - \frac, for complex numbers with large modulus (, z, \rightarrow\infty) in the Circular sector, sector , \arg z, 0. The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning Gamma, double-gamma). Gamma. Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: :\log \Gamma(z+1)=\log(z)+\log \Gamma(z), Differentiating both sides with respect to gives: :\psi(z+1)=\psi(z)+\frac Since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particular Values Of The Gamma Function
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational number, rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations. Integers and half-integers For positive integer arguments, the gamma function coincides with the factorial. That is, :\Gamma(n) = (n-1)!, and hence :\begin \Gamma(1) &= 1, \\ \Gamma(2) &= 1, \\ \Gamma(3) &= 2, \\ \Gamma(4) &= 6, \\ \Gamma(5) &= 24, \end and so on. For non-positive integers, the gamma function is not defined. For positive half-integers \frac where k\in 2\mathbb^*+1 is an odd integer greater or equal 3, the function values are given exactly by :\Gamma \left (\tfrac \right) = \sqrt \pi \frac\,, or equivalently, for non-negative integer values of : :\begin \Gamma\left(\tfr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... is Vadim Ponomarenko ( San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function (mathematics), function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function. Formally, the Barnes ''G''-function is defined in the following Weierstrass product form: : G(1+z)=(2\pi)^ \exp\left(- \frac \right) \, \prod_^\infty \left\ where \, \gamma is the Euler–Mascheroni constant, exponential function, exp(''x'') = ''e''''x'' is the exponential function, and Π denotes multiplication (capital pi notation). The integral representation, which may be deduced from the relation to the double gamma function, is : \log G(1+z) = \frac\log(2\pi) +\int_0^\infty\frac\left[\frac +\frace^ -\frac\right] As an entire function, ''G'' is of order two, and of infinite type. This can be deduced from the asymptoti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
