A261025 -id:A261025 - cp's OEIS frontend

A261024 Decimal expansion of Cl_2(2*Pi/3), where Cl_2 is the Clausen function of order 2.

6, 7, 6, 6, 2, 7, 7, 3, 7, 6, 0, 6, 4, 3, 5, 7, 5, 0, 0, 1, 4, 1, 3, 5, 0, 3, 6, 1, 8, 3, 0, 1, 3, 5, 2, 3, 9, 6, 1, 1, 2, 6, 2, 0, 5, 0, 2, 0, 1, 9, 9, 8, 6, 1, 3, 4, 4, 9, 9, 2, 7, 3, 7, 8, 5, 1, 0, 6, 4, 9, 8, 4, 1, 7, 2, 1, 6, 2, 6, 8, 1, 4, 2, 4, 3, 1, 3, 5, 6, 9, 4, 8, 5, 5, 0, 4, 4, 6, 3, 2, 9, 7, 2, 4, 1

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.676627737606435750014135036183013523961126205020199861344992737851...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function.
Eric Weisstein's MathWorld, Clausen's Integral.
Eric Weisstein's MathWorld, Barnes G-Function.
Wikipedia, Clausen function.
Wikipedia, Barnes G-function.

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).

Cf. A252798, A252799, A263416, A263417.

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[2*Pi/3] // Re, 10, 105] // First

clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
clausen(2, 2*Pi/3) \\ Gheorghe Coserea, Sep 30 2018

Equals 2*Pi*log(G(2/3)/G(1/3)) - 2*Pi*LogGamma(1/3) + (2*Pi/3)*log(2*Pi/sqrt(3)), where G is the Barnes G function.

A261027 Decimal expansion of Cl_2(Pi/6), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

8, 6, 4, 3, 7, 9, 1, 3, 1, 0, 5, 3, 8, 9, 2, 7, 4, 9, 6, 2, 5, 0, 3, 6, 3, 1, 5, 1, 9, 0, 2, 1, 9, 4, 7, 8, 6, 6, 8, 1, 8, 8, 5, 7, 6, 4, 0, 3, 6, 8, 9, 7, 0, 4, 1, 8, 2, 0, 3, 7, 6, 8, 9, 7, 7, 5, 3, 2, 4, 7, 1, 5, 5, 8, 2, 0, 6, 4, 1, 8, 5, 1, 8, 7, 0, 2, 1, 9, 3, 0, 5, 0, 7, 8, 0, 7, 5, 7, 7, 9, 0, 2, 1, 8

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.8643791310538927496250363151902194786681885764036897041820...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261028 (Cl_2(5*Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[Pi/6] // Re, 10, 104] // First

Equals 2*Pi*log(G(11/12)/G(1/12)) - 2*Pi*LogGamma(1/12) + (Pi/6) * log(2*Pi*sqrt(2)/(sqrt(3)-1)), where G is the Barnes G function.

A261028 Decimal expansion of Cl_2(5*Pi/6), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

3, 5, 6, 9, 0, 8, 3, 2, 7, 8, 4, 9, 0, 6, 5, 9, 3, 7, 1, 1, 4, 4, 3, 5, 0, 3, 8, 0, 5, 2, 9, 5, 9, 3, 3, 5, 6, 9, 7, 3, 4, 3, 9, 2, 2, 6, 3, 8, 5, 3, 9, 8, 0, 8, 1, 7, 3, 2, 9, 3, 1, 3, 6, 3, 8, 7, 0, 2, 5, 9, 7, 7, 4, 5, 2, 9, 4, 2, 1, 7, 4, 1, 1, 8, 7, 8, 6, 7, 5, 3, 3, 9, 3, 6, 7, 9, 2, 3, 0, 4, 2, 9, 2, 5

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.356908327849065937114435038052959335697343922638539808173...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[5*Pi/6] // Re, 10, 104] // First

Equals 2*Pi*log(G(7/12)/G(5/12)) - 2*Pi*LogGamma(5/12) + (5*Pi/6) * log(2*Pi*sqrt(2)/(sqrt(3)+1)), where G is the Barnes G function.

A261026 Decimal expansion of Cl_2(3*Pi/4), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

5, 2, 3, 8, 8, 9, 3, 5, 3, 9, 6, 1, 5, 9, 3, 8, 4, 9, 1, 9, 0, 6, 2, 2, 7, 8, 5, 5, 9, 4, 0, 0, 3, 6, 1, 1, 7, 4, 3, 7, 1, 6, 2, 8, 4, 5, 1, 9, 8, 9, 4, 3, 9, 4, 4, 4, 3, 3, 6, 4, 0, 7, 4, 8, 4, 2, 2, 7, 4, 1, 5, 5, 1, 6, 1, 6, 4, 2, 2, 5, 1, 4, 8, 5, 2, 2, 4, 5, 4, 6, 4, 2, 1, 3, 3, 0, 1, 7, 0, 9, 9, 9, 7, 0, 9

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.52388935396159384919062278559400361174371628451989439444336407484...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261025 (Cl_2(Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[3*Pi/4] // Re, 10, 105] // First

Equals 2*Pi*log(G(5/8)/G(3/8)) - 2*Pi*LogGamma(3/8) + (3*Pi/4) * log(2*Pi/sqrt(2+sqrt(2))), where G is the Barnes G function.

A340826 Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

9, 2, 3, 7, 5, 5, 1, 6, 8, 1, 0, 0, 5, 3, 5, 3, 0, 8, 7, 1, 1, 9, 8, 6, 0, 2, 9, 7, 9, 3, 0, 2, 4, 3, 5, 3, 9, 6, 6, 2, 7, 9, 0, 0, 6, 4, 1, 2, 5, 1, 7, 2, 5, 1, 7, 0, 7, 7, 1, 2, 8, 4, 8, 3, 2, 5, 1, 5, 0, 9, 8, 3, 3, 2, 5, 3, 0, 9, 7, 5, 7, 2, 8, 7, 2, 8, 3, 2, 2, 1, 8, 0, 1, 1, 2, 2, 5, 9, 9, 9, 6, 2, 6, 3, 5

Offset: 0

Artur Jasinski, Jan 23 2021

			0.9237551681005353087119860297930...

For links see A261024.

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)), A340628, A340629.

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Re[Cl2[Pi/5]], 10, 105] // First
N[Pi*(ArcCsch[2] + Log[2*Pi*BarnesG[9/10]^10 / BarnesG[11/10]^10])/5, 120] (* Vaclav Kotesovec, Jan 23 2021 *)

A = Cl_2(Pi/5).
B = Cl_2(2*Pi/5).
C = Cl_2(3*Pi/5).
D = Cl_2(4*Pi/5).
4*(A^2 + C^2) = 5*(B^2 + D^2).
B = 2*A - 2*D.
D = 2*B - 2*C.
2*C = 4*A - 5*D.
B = -D + sqrt(A*(2*C+D)+D^2).
B^2 + D^2 = 4*Pi^4/(325*A340628^2).
B^2 + D^2 = (13/1125)*A340629^2*Pi^4.
Equals Pi*(2*log(G(9/10) / G(11/10)) + log(Pi*(1+sqrt(5)))/5), where G is the Barnes G-function. - Vaclav Kotesovec, Jan 23 2021

cp's OEIS Frontend

A261024 Decimal expansion of Cl_2(2*Pi/3), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A261027 Decimal expansion of Cl_2(Pi/6), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A261028 Decimal expansion of Cl_2(5*Pi/6), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A261026 Decimal expansion of Cl_2(3*Pi/4), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A340826 Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula