"Log sine function" redirects here. For the historically used compound function, see logarithmic sine.
"Log cosine function" redirects here. For the historically used compound function, see logarithmic cosine.
Graph of the Clausen function Cl2(θ)
In mathematics, the Clausen function, introduced by Thomas Clausen (1832), 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:
In the range the sine function inside the absolute value sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has the Fourier series representation:
The Clausen functions, as a class of functions, feature extensively in many areas of modern mathematical research, particularly in relation to the evaluation of many classes of logarithmic and polylogarithmic integrals, both definite and indefinite. They also have numerous applications with regard to the summation of hypergeometric series, summations involving the inverse of the central binomial coefficient, sums of the polygamma function, and Dirichlet L-series.
In mathematics, the Clausenfunction, introduced by Thomas Clausen (1832), is a transcendental, special function of a single variable. It can variously...
A similar result holds for the trigamma function but the circular functions are replaced by Clausen'sfunction. Namely, ψ 1 ( p q ) = π 2 2 sin 2 ( π...
Clausenfunction Complete Fermi–Dirac integral, an alternate form of the polylogarithm. Dilogarithm Incomplete Fermi–Dirac integral Kummer's function...
{z^{k}}{k^{s}}}} which coincides with the Riemann zeta function when z = 1. The Clausenfunction Cls(θ) can be chosen as the real or imaginary part of...
Clausen is a Danish patronymic surname, literally meaning child of Claus, Claus being a German form of the Greek Νικόλαος, Nikolaos, (cf. Nicholas), used...
This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks...
} The function D(z) is sometimes called the Bloch-Wigner function. Lobachevsky's function and Clausen'sfunction are closely related functions. William...
Tim Clausen (born 12 March 1969) is a structural biologist and a senior scientist at the Research Institute of Molecular Pathology (IMP) in Vienna, Austria...
_{2}\left(\varphi \right)} is the Clausenfunction. Similarly, Catalan's constant may also be expressed in terms of the Clausenfunction, K = Cl 2 ( π 2 ) = ∑...
polylogarithm with pure imaginary μ may be expressed in terms of the Clausenfunctions Cis(θ) and Sis(θ), and vice versa (Lewin 1958, Ch. VII § 1.4; Abramowitz...
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied...
complex numbers leads to the following paradox, discovered in 1827 by Clausen: For any integer n, we have: e 1 + 2 π i n = e 1 e 2 π i n = e ⋅ 1 = e...
coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over...
{6}}.\end{cases}}} The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt and Thomas Clausen independently in 1840. The theorem...
an ideal tetrahedron can be expressed in terms of the Clausenfunction or Lobachevsky function of its dihedral angles, and the volume of an arbitrary...
maintenance of fluid and electrolyte balance. Potassium is necessary for the function of all living cells, and is thus present in all plant and animal tissues...
problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain the optimal solution. It...
ETags to Reduce Bandwidth & Workload with Spring & Hibernate ETag in HTTP/1.1 specification Concerning Etags and Datestamps by Lars R. Clausen (2004)...
squares Riemann zeta function Basel problem on ζ(2) Hurwitz zeta function Bernoulli number Agoh–Giuga conjecture Von Staudt–Clausen theorem Dirichlet series...
ISSN 1088-6222. PMID 21186707. Chan, T. C.; Vilke, G. M.; Neuman, T.; Clausen, J. L. (November 1997). "Restraint position and positional asphyxia". Annals...
ISBN 978-0-87893-243-6. Weschler CJ, Langer S, Fischer A, Bekö G, Toftum J, Clausen G (May 2011). "Squalene and cholesterol in dust from danish homes and daycare...
Global Information
