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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 integral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 x. 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] |
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 or memory used by some algorithms (e.g., "it has polylogarithmic order"). 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. This observation is the basis for the soft O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ... . References * Mathematical analysis Polynomials Analysis of algorithms {{comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lerch Transcendent
In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, 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 the function in 1887. Definition The Lerch zeta function is given by :L(\lambda, s, \alpha) = \sum_^\infty \frac . A related function, the Lerch transcendent, is given by :\Phi(z, s, \alpha) = \sum_^\infty \frac . The two are related, as :\,\Phi(e^, s,\alpha)=L(\lambda, s, \alpha). Integral representations The Lerch transcendent has an integral representation: : \Phi(z,s,a)=\frac\int_0^\infty \frac\,dt The proof is based on using the integral definition of the Gamma function to write :\Phi(z,s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty t^s z^n e^ \frac and then interchanging the sum and integral. The resulting integral representation converges for z \in \Complex \s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned .... 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_^ (-)^ \;\frac \;\operatorname_ (z) \;\;+\;\; \frac \;\left \Lambda_n(-1) - \Lambda_n(-z) \right References *. Special functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indefinite Integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a 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 and acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 an infinite series representation in terms of which it is initially defined 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'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilogarithm
In mathematics, Spence's function, or dilogarithm, 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-Winger function. Lobachevsky's function and Clausen's function are closely related functions. William Spence, after whom the function was named by early writers in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to be any complex number w for which e^w = z.Ahlfors, Section 3.4.Sarason, Section IV.9. Such a number w is denoted by \log z. If z is given in polar form as z = re^, where r and \theta are real numbers with r>0, then \ln r + i \theta is one logarithm of z, and all the complex logarithms of z are exactly the numbers of the form \ln r + i\left(\theta + 2\pi k\right) for integers ''k''. These logarithms are equally spaced along a vertical line in the complex plane. * A complex-valued function \log \colon U \to \mathbb, defined on some subset U of the set \mathbb^* of nonzero complex numbers, satisfying e^ = z for all z in U. Such complex logarithm functions are analogous to the real logarithm function \ln \colon \mathbb_ \to \mathbb, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Branch
In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are used in the definition of many inverse trigonometric functions, such as the selection either to define that :\arcsin:1,+1rightarrow\left \frac,\frac\right/math> or that :\arccos:1,+1rightarrow ,\pi/math>. Exponentiation to fractional powers A more familiar principal branch function, limited to real numbers, is that of a positive real number raised to the power of . For example, take the relation , where is any positive real number. This relation can be satisfied by any value of equal to a square root of (either positive or negative). By convention, is used to denote the positive square root of . In this instance, the positive square root function is taken as the principal branch of the multi-valued relation . Complex logarithms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 products. Suppose that ''A'' is a set with a function ''w'': ''A'' → N assigning a weight to each of the elements of ''A'', and suppose additionally that the fibre over any natural number under that weight is a finite set. (We call such an arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
