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Dilogarithm
In mathematics, the dilogarithm (or Spence's function), denoted as , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: :\operatorname_2(z) = -\int_0^z\, du \textz \in \Complex and its reflection. For , an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane): :\operatorname_2(z) = \sum_^\infty . Alternatively, the dilogarithm function is sometimes defined as :\int_^ \frac dt = \operatorname_2(1-v). In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex whose vertices have cross ratio has hyperbolic volume :D(z) = \operatorname \operatorname_2(z) + \arg(1-z) \log, z, . The function is sometimes called the Bloch-Wigner function. Lobachevsky's function and Clausen's function are closely related functions. William Spence, after whom the function was named by early write ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Quantum Dilogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula : \phi(x)\equiv(x;q)_\infty=\prod_^\infty (1-xq^n),\quad , q, 0. References * * * * * * * External links * {{nlab, id=quantum+dilogarithm, title=quantum dilogarithm Special functions Q-analogs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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William Spence (mathematician)
William Spence (born 31 July 1777 in Greenock, Scotland – died 22 May 1815 in Glasgow, Scotland) was a Scottish mathematician who published works on the fields of logarithmic functions, algebraic equations and their relation to integral and differential calculus respectively. Early life, family, and personal life Spence was the second son to Ninian Spence and his wife Sarah Townsend. Ninian Spence ran a coppersmith business, and the Spence family were a prominent family in Greenock at the time. From an early age, Spence was characterised as having a docile and reasonable nature, with him being mature for his age. At school he formed a life-long friendship with John Galt, who documented much of his life and his works posthumously. Despite having received a formal education until he was a teenager, Spence never attended university, instead he moved to Glasgow where he lodged with a friend of his fathers, learning the skills of a manufacturer. Two years after his father's de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Markstein Number
In combustion engineering and explosion studies, the Markstein number (named after George H. Markstein who first proposed the notion in 1951) characterizes the effect of local heat release of a propagating flame on variations in the surface topology along the flame and the associated local flame front curvature. There are two dimensionless Markstein numbers:Clavin, Paul, and Geoff Searby. Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation Fronts and Explosion of Stars. Cambridge University Press, 2016. one is the curvature Markstein number and the other is the flow-strain Markstein number. They are defined as: :\mathcal_c = \frac, \quad \mathcal_s = \frac where \mathcal_c is the curvature Markstein length, \mathcal_s is the flow-strain Markstein length and \delta_L is the characteristic laminar flame thickness. The larger the Markstein length, the greater the effect of curvature on localised burning velocity. George H. Markstein (1911—2011) showed that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Reflection Formula
In mathematics, a reflection formula or reflection relation for a function is a relationship between and . It is a special case of a functional equation. It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae. Reflection formulae are useful for numerical computation of special functions. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the complex plane) can be employed for all arguments. Known formulae The even and odd functions satisfy by definition simple reflection relations around . For all even functions, f(-x) = f(x), and for all odd functions, f(-x) = -f(x). A famous relationship is Euler's reflection formula \Gamma(z)\Gamma(1-z) = \frac, \qquad z \not\in \mathbb Z for the gamma function \Gamma(z), due to Leonhard Euler. There is also a reflection formula for the general -th order polygamma function , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Radiative Correction
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian. For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Leonard James Rogers
Leonard James Rogers FRS (30 March 1862 – 12 September 1933) was a British mathematician who was the first to discover the Rogers–Ramanujan identity and Hölder's inequality, and who introduced Rogers polynomials. The Rogers–Szegő polynomials are named after him. Early life and education Rogers was born in Oxford, the second son of James Edwin Thorold Rogers and his second wife Anne Reynolds, and brother of Annie Rogers. He matriculated at Balliol College, Oxford, graduating BA and BMus in 1884 and MA in 1887. Academic career Rogers became lecturer in mathematics at Wadham College, Oxford in 1885. In 1888 Rogers was appointed Professor of Mathematics at the Yorkshire College, by then a constituent college of the Victoria University. The Yorkshire College became the University of Leeds in 1904. In 1919 he retired because of poor health. Rogers worked initially on reciprocants in the theory of differential invariants, and then moved into the area of special func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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John Galt (novelist)
John Galt (; 2 May 1779 – 11 April 1839) was a Scottish novelist, entrepreneur, and political and social commentator. Galt has been called the first political novelist in the English language, due to being the first novelist to deal with issues of the Industrial Revolution. Galt was the first superintendent of the Canada Company (1826–1829). The company was formed to populate a part of what is now Southern Ontario (then known as Upper Canada) in the first half of the 19th century; it was later called "the most important single attempt at settlement in Canadian history". In 1829, Galt was recalled to Great Britain for mismanagement of the Canada Company (particularly incompetent bookkeeping), and was later jailed for failing to pay his son's tuition. Galt's ''Autobiography'', published in London in 1833, includes a discussion of his life and work in Upper Canada. He was the father of Sir Alexander Tilloch Galt of Montreal, Quebec, one of the leading Fathers of Confederation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Lobachevsky's Function
In mathematics, the Clausen function, introduced by , is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred to as ''the'' Clausen function, despite being but one of a class of many – is given by the integral: :\operatorname_2(\varphi)=-\int_0^\varphi \log\left, 2\sin\frac \\, dx: In the range 0 :\operatorname_2\left(-\frac+2m\pi \right) =-1.01494160 \ldots The following properties are immediate consequences of the series definition: :\operatorname_2(\theta+2m\pi) = \operatorname_2(\theta) :\operatorname_2(-\theta) = -\operatorname_2(\theta) See . General definition More generally, one defines the two generalized Clausen funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Clausen's Function
In mathematics, the Clausen function, introduced by , is a Transcendental number, transcendental, special Function (mathematics), function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred to as ''the'' Clausen function, despite being but one of a class of many – is given by the integral: :\operatorname_2(\varphi)=-\int_0^\varphi \log\left, 2\sin\frac \\, dx: In the range 0 < \varphi < 2\pi\, the sine function inside the absolute value sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has the Fourier series representation: : [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |