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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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
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] [Amazon] |
