Particular Values Of The Gamma Function
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational number, rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations. Integers and half-integers For positive integer arguments, the gamma function coincides with the factorial. That is, :\Gamma(n) = (n-1)!, and hence :\begin \Gamma(1) &= 1, \\ \Gamma(2) &= 1, \\ \Gamma(3) &= 2, \\ \Gamma(4) &= 6, \\ \Gamma(5) &= 24, \end and so on. For non-positive integers, the gamma function is not defined. For positive half-integers \frac where k\in 2\mathbb^*+1 is an odd integer greater or equal 3, the function values are given exactly by :\Gamma \left (\tfrac \right) = \sqrt \pi \frac\,, or equivalently, for non-negative integer values of : :\begin \Gamma\left(\tfr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complete Elliptic Integral Of The First Kind
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Carlo de' Toschi di Fagnano, Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function (mathematics), function which can be expressed in the form f(x) = \int_^ R \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate Integration by reduction formulae, reduction formula, every elliptic integral can be brought into a form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Local Minimum
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' extrema) or on the entire domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Modulus Squared
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 (caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is '' quadratic''. The square of an integer may also be called a ''square number'' or a ''perfect square''. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Fractional Part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than , called ''floor'' of or \lfloor x\rfloor. Then, the fractional part can be formulated as a difference: :\operatorname (x)=x - \lfloor x \rfloor,\; x > 0. The fractional part of logarithms, specifically, is also known as the mantissa; by contrast with the mantissa, the integral part of a logarithm is called its ''characteristic''. The word ''mantissa'' was introduced by Henry Briggs. For a positive number written in a conventional positional numeral system (such as binary or decimal), its fractional part hence corresponds to the digits appearing after the radix point, such as the decimal point in English. The result is a real number in the half-open interval x, -\lfloor , x, \rfloor , or by the : [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function (mathematics), function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function. Formally, the Barnes ''G''-function is defined in the following Weierstrass product form: : G(1+z)=(2\pi)^ \exp\left(- \frac \right) \, \prod_^\infty \left\ where \, \gamma is the Euler–Mascheroni constant, exponential function, exp(''x'') = ''e''''x'' is the exponential function, and Π denotes multiplication (capital pi notation). The integral representation, which may be deduced from the relation to the double gamma function, is : \log G(1+z) = \frac\log(2\pi) +\int_0^\infty\frac\left[\frac +\frace^ -\frac\right] As an entire function, ''G'' is of order two, and of infinite type. This can be deduced from the asymptoti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Imaginary Unit
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is Imaginary numbers are an important mathematical concept; they extend the real number system \mathbb to the complex number system \mathbb, in which at least one Root of a function, root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term ''imaginary'' is used because there is no real number having a negative square (algebra), square. There are two complex square roots of and , just as there are two complex square roots of every real number other than zero (which has one multiple root, double square root). In contexts in which use of the letter is ambiguous or problematic, the le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Carl Johan Malmsten
Carl Johan Malmsten (April 9, 1814, in Uddetorp, Skara County, Sweden – February 11, 1886, in Uppsala, Sweden) was a Swedish mathematician and politician. He is notable for early research into the theory of functions of a complex variable, for the evaluation of several important logarithmic integrals and series, for his studies in the theory of Zeta-function related series and integrals, as well as for helping Mittag-Leffler start the journal ''Acta Mathematica''. Malmsten became Docent in 1840, and then, Professor of mathematics at the Uppsala University in 1842. He was elected a member of the Royal Swedish Academy of Sciences in 1844. He was also a minister without portfolio in 1859–1866 and Governor of Skaraborg County in 1866–1879. Main contributions Usually, Malmsten is known for his earlier works in complex analysis. However, he also greatly contributed in other branches of mathematics, but his results were undeservedly forgotten and many of them were erroneously at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Theta Function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Catalan's Constant
In mathematics, Catalan's constant , is the alternating sum of the reciprocals of the odd square numbers, being defined by: : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : 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 theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form n^2+1 accordin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glaisher–Kinkelin Constant
In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to special functions like the -function and the Barnes -function. The constant also appears in a number of sums and integrals, especially those involving the gamma function and the Riemann zeta function. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin. Its approximate value is: : = ... . Glaisher's constant plays a role both in mathematics and in physics. It appears when giving a closed form expression for Porter's constant, when estimating the efficiency of the Euclidean algorithm. It also is connected to solutions of Painlevé differential equations and the Gaudin model. Definition The Glaisher–Kinkelin constant can be defined via the following limit: :A=\lim_ \frac where H(n) is the hyperfactorial: H(n)= \prod_^ i^i = 1^1\cdot 2^2\cdot 3^3 \cdot \cdot n^nAn analogous limit, presenting a similarit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |