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Trigamma Function
In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by : \psi_1(z) = \frac \ln\Gamma(z). It follows from this definition that : \psi_1(z) = \frac \psi(z) where is the digamma function. It may also be defined as the sum of the series : \psi_1(z) = \sum_^\frac, making it a special case of the Hurwitz zeta function : \psi_1(z) = \zeta(2,z). Note that the last two formulas are valid when is not a natural number. Calculation A double integral representation, as an alternative to the ones given above, may be derived from the series representation: : \psi_1(z) = \int_0^1\!\!\int_0^x\frac\,dy\,dx using the formula for the sum of a geometric series. Integration over yields: : \psi_1(z) = -\int_0^1\frac\,dx An asymptotic expansion as a Laurent series is : \psi_1(z) = \frac + \frac + \sum_^\frac = \sum_^\frac if we have chosen , i.e. the Bernoulli numbers of the second kind. Recurrence and reflection formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygamma Function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) = \psi(z) = \frac holds where is the digamma function and is the gamma function. They are holomorphic on \mathbb \backslash\mathbb_. At all the nonpositive integers these polygamma functions have a pole of order . The function is sometimes called the trigamma function. Integral representation When and , the polygamma function equals :\begin \psi^(z) &= (-1)^\int_0^\infty \frac\,\mathrmt \\ &= -\int_0^1 \frac(\ln t)^m\,\mathrmt\\ &= (-1)^m!\zeta(m+1,z) \end where \zeta(s,q) is the Hurwitz zeta function. This expresses the polygamma function as the Laplace transform of . It follows from Bernstein's theorem on monotone functions that, for and real and non-negative, is a completely monotone function. Setting in the above formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strictly concave on (0,\infty). The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning double-gamma). Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the derivative with respect to gives: :\Gamma'(z+1)=z\Gamma'(z)+\Gamma(z) \, Dividing by or the equivalent gives: :\frac=\frac+\frac or: :\psi(z+1)=\psi(z)+\frac Since the harmonic numbers are defined for positive integers as :H_n=\sum_^n \frac 1 k, the digamma function is related to them by :\psi(n)=H_-\gamma, where and is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values : \psi \left(n+\tfrac12\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clausen 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 fu ... [...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 number, irrational, let alone transcendental number, 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 polyhedron, 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, spannin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Formula
In mathematics, a reflection formula or reflection relation for a function ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a functional equation, and it is very common in the literature to use the term "functional equation" when "reflection formula" is meant. Reflection formulas 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 ''a'' = 0. 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 reflect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.. Definition The Laurent series for a complex function f(z) about a point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where a_n and c are constants, with a_n defined by a line integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic (analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
