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Inverse Tangent Integral
The inverse tangent integral is a special function, defined by: :\operatorname_2(x) = \int_0^x \frac \, dt Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function. Definition The inverse tangent integral is defined by: :\operatorname_2(x) = \int_0^x \frac \, dt The arctangent is taken to be the principal branch; that is, −/2 0). This can be proven by differentiating and using the identity \arctan(t) + \arctan(1/t) = \pi/2. The special value Ti2(1) is Catalan's constant 1 - \frac + \frac - \frac + \cdots \approx 0.915966. Generalizations Similar to the polylogarithm \operatorname_n(z) = \sum_^\infty \frac, the function :\operatorname_n(x) = x - \frac + \frac - \frac + \cdots is defined analogously. This satisfies the recurrence relation: :\operatorname_n(x) = \int_0^x \frac \, dt Relation to other special functions The inverse tangent integral is related to the Legendre chi function \chi_2(x) = x + \frac + \frac + \c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewe ... [...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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ... 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) = \opera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arctangent
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent (trigonometry), tangent, cotangent, secant (trigonometry), secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. Notation Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of ''θ'' radians will correspond to an arc whose length is ''rθ'', where ''r'' is ... [...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] |
Absolutely Convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said to converge absolutely if \textstyle\sum_^\infty \left, a_n\ = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x)\,dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty , f(x), dx = L. Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess - a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Function
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is an even function if ''n'' is an even integer, and it is an odd function if ''n'' is an odd integer. Definition and examples Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on. The gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan's Constant
In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is irrational, let alone transcendental. has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven". Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865. Uses In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link. It is 1/8 of the volume of the complement of the Borromean rings. In combinatorics and statistical mechanics, it arises in connection with counting domino tilings, spanning trees, and Hamiltonian cycles of grid graphs. In number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Legendre Chi Function
In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by \chi_\nu(z) = \sum_^\infty \frac. As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially expressible in terms of the polylogarithm as \chi_\nu(z) = \frac\left operatorname_\nu(z) - \operatorname_\nu(-z)\right The Legendre chi function appears as the discrete Fourier transform, with respect to the order ν, of the 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 ..., and also of the Euler polynomials, with the explicit relationships given in those articles. The Legendre chi function is a special case of the Lerch transcendent, and is given by \chi_\nu(z)=2^z\,\Phi (z^2,\nu,1/2). Identities \chi_2(x) + \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Hyperbolic Tangent
In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The size of the hyperbolic angle is equal to the area of the corresponding hyperbolic sector of the hyperbola , or twice the area of the corresponding sector of the unit hyperbola , just as a circular angle is twice the area of the circular sector of the unit circle. Some authors have called inverse hyperbolic functions "area functions" to realize the hyperbolic angles. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. It also occurs in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat tra ... [...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 \set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
