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Clausen function

In mathematics, the Clausen function, introduced by Thomas Clausen (1832), is a transcendental, special function of a single variable. It can 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…

Applications, Overview & Integral evaluations involving the direct function

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Overview

MathematicsThomasClausen1832TranscendentalSpecial function
21 related topics

Integral evaluations involving the direct function

Mathematical constantsZeta functionTrigonometryIntegration by parts
4 related topics

General definition

Complex planeAnalytic continuationJames Whitbread Lee Glaisher
3 related topics

Relation to Dirichlet L-functions

Cyclic groupHurwitz zeta functionDirichlet L-functions
3 related topics

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Overview

Basic properties

General definition

Relation to the Bernoulli polynomials

Duplication formula

Derivatives of general-order Clausen functions

Relation to the Barnes' G-function

Relation to the polylogarithm

Kummer's relation

Relation to the Lobachevsky function

Relation to Dirichlet L-functions

Series acceleration

Special values

Integrals of the direct function

Integral evaluations involving the direct function

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Clausen function

Nodes65
Edges64
Triples123
Avg. degree1.97
Density0.030769
Components1

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Clausen function

Top relations

related to References · 81
Clausen function → Abramowitz, Adamchik, American Mathematical Society, Appl, Applied Mathematics Series, Approx, Archived, Armin, Barnes Function, Bibcode, Borwein, Bradley, C99, Chapter, Chebyshev, Christopher, Cl2, Clausen, Clausen's, Commerce
related to Relation to the Barnes' G-function · 6
Clausen function → Barnes G-function, Cl, Clausen, Euler, For, Gamma
related to Derivatives of general-order Clausen functions · 5
Clausen function → Cl, Clausen, Direct, Fourier, Sl
related to Relation to the polygamma function · 5
Clausen function → Cl, Clausen, Indeed, One, The Clausen
related to Relation to the polylogarithm · 5
Clausen function → Cl, Im, Li, Re, The Clausen
related to Integral evaluations involving the direct function · 4
Clausen function → Catalan's, Clausen, Fourier, The
related to Integrals of the direct function · 4
Clausen function → Cl, Clausen, Sl, The
related to Relation to the Bernoulli polynomials · 4
Clausen function → Bernoulli, Fourier, The SL-type Clausen, This
related to Generalized special values · 3
Clausen function → Cl, Clausen, Some
related to Basic properties · 2
Clausen function → Cl, The Clausen

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Important terminology

displaystyle function clausen operatorname cl frac theta pi sin left right series functions integral zeta log sum 2m -1 int

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Clausen functionrelated to Basic propertiesThe Clausen0.60section
Clausen functionrelated to Basic propertiesCl0.60section
Clausen functionrelated to Derivatives of general-order Clausen functionsDirect0.60section
Clausen functionrelated to Derivatives of general-order Clausen functionsFourier0.60section
Clausen functionrelated to Derivatives of general-order Clausen functionsClausen0.60section
Clausen functionrelated to Derivatives of general-order Clausen functionsCl0.60section
Clausen functionrelated to Derivatives of general-order Clausen functionsSl0.60section
Clausen functionrelated to General definitionMore0.60section
Clausen functionrelated to General definitionClausen0.60section
Clausen functionrelated to Generalized special valuesSome0.60section
Clausen functionrelated to Generalized special valuesClausen0.60section
Clausen functionrelated to Generalized special valuesCl0.60section

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