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]   |
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Polylogarithmic Function
In mathematics, a polylogarithmic function in is a polynomial in the logarithm of , : a_k (\log n)^k + a_ (\log n)^ + \cdots + a_1(\log n) + a_0. The notation is often used as a shorthand for , analogous to for . In computer science, polylogarithmic functions occur as the order of time for some data structure In computer science, a data structure is a data organization and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the relationships amo ... operations. Additionally, the exponential function of a polylogarithmic function produces a function with quasi-polynomial growth, and algorithms with this as their time complexity are said to take quasi-polynomial time. All polylogarithmic functions of are for every exponent (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Lerch Transcendent
In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887. The Lerch transcendent, is given by: :\Phi(z, s, \alpha) = \sum_^\infty \frac . It only converges for any real number \alpha > 0, where , z, 1, and , z, = 1. Special cases The Lerch transcendent is related to and generalizes various special functions. The Lerch zeta function is given by: :L(\lambda, s, \alpha) = \sum_^\infty \frac =\Phi(e^, s,\alpha) The Hurwitz zeta function is the special case :\zeta(s,\alpha) = \sum_^\infty \frac = \Phi(1,s,\alpha) The polylogarithm is another special case: :\textrm_s(z) = \sum_^\infty \frac =z\Phi(z,s,1) The Riemann zeta function is a special case of both of the above: :\zeta(s) =\sum_^\infty \frac = \Phi(1,s,1) The Dirichlet eta function: :\eta(s) = \sum_^\infty \frac = \Phi(-1,s,1) The Diric ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Multiplication Formula
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises. Finite characteristic The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of terms are added or multiplied. The finite form typically occurs only for the gamma and related functions, for which the identity follows from a p-adic relation over a finite field. For example, the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of complex m ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Kummer's Function
In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h .... Kummer's function is defined by :\Lambda_n(z)=\int_0^z \frac\;dt. The duplication formula is :\Lambda_n(z)+\Lambda_n(-z)= 2^\Lambda_n(-z^2). Compare this to the duplication formula for the polylogarithm: :\operatorname_n(z)+\operatorname_n(-z)= 2^\operatorname_n(z^2). An explicit link to the polylogarithm is given by :\operatorname_n(z)=\operatorname_n(1)\;\;+\;\; \sum_^ (-1)^ \;\frac \;\operatorname_ (z) \;\;+\;\; \frac \;\left \Lambda_n(-1) - \Lambda_n(-z) \right References *. Special functio ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Duplication Formula
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises. Finite characteristic The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of terms are added or multiplied. The finite form typically occurs only for the gamma and related functions, for which the identity follows from a p-adic relation over a finite field. For example, the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of complex m ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Offset Logarithmic Integral
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value . Integral representation The logarithmic integral has an integral representation defined for all positive real numbers ≠ 1 by the definite integral : \operatorname(x) = \int_0^x \frac. Here, denotes the natural logarithm. The function has a singularity at , and the integral for is interpreted as a Cauchy principal value, : \operatorname(x) = \lim_ \left( \int_0^ \frac + \int_^x \frac \right). Offset logarithmic integral The offset logarithmic integral or Eulerian logarithmic integral is defined as : \operatorname(x) = \int_2^x \frac = \operatorname(x) - \operatorname(2). ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Hurwitz Zeta Function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by :\zeta(s,a) = \sum_^\infty \frac. This series is absolutely convergent for the given values of and and can be extended to a meromorphic function defined for all . The Riemann zeta function is . The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. Integral representation The Hurwitz zeta function has an integral representation :\zeta(s,a) = \frac \int_0^\infty \frac dx for \operatorname(s)>1 and \operatorname(a)>0. (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing :\zeta(s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty y^s e^ \frac and then interchanging the sum and integral. The integral representation above can be converted to a contour integral representation :\zeta(s,a) = -\Gamma(1-s)\frac \int_C \frac dz wh ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Dirichlet Series
In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ''ζ(s)'' is the Dirichlet series of the constant unit function ''u(n)'', namely: \zeta(s) = \sum_^\infty \frac = \sum_^\infty \frac = D(u, s), where ''D(u, s)'' denotes the Dirichlet series of ''u(n)''. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet. Combinatorial importance Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian product ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Indefinite Integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically as . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called ''differentiation'', which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as and . Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Mercator Series
In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac+\frac-\frac+\cdots In summation notation, :\ln(1+x)=\sum_^\infty \frac x^n. The series converges to the natural logarithm (shifted by 1) whenever -1 History The series was discovered independently by Johannes Hudde (1656) and (1665) but neither published the result. Nicholas Mercator also independently discovered it, and included values of the series for small values in his 1668 treatise ''Logarithmotechnia''; the general series was included in[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |