A261028 -id:A261028 - 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.

A261025 Decimal expansion of Cl_2(Pi/4), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.9818721510502033567179245430601956671307909716607304615766131...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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)), A261026 (Cl_2(3*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[Pi/4] // Re, 10, 105] // First

Equals 2*Pi*log(G(7/8)/G(1/8)) - 2*Pi*LogGamma(1/8) + (Pi/4) * log(2*Pi/sqrt(2-sqrt(2))), 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.

A244843 Decimal expansion of the integral of log(2+x^2+y^2)/((1+x^2)*(1+y^2)) dx dy over the square [0,1]x[0,1].

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 07 2014

The computation of this integral is given by Bailey & Borwein as an example of the use of CAS packages (and additional tools) to simplify large symbolic expressions.

			0.56959615818361450623645553672717469010787612682122878368281840812485230025...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Bailey and J. M. Borwein, Experimental computation as an ontological game changer, 2014, see p. 5.
D. H. Bailey, J. M. Borwein and A. D. Kaiser, Automated Simplification of Large Symbolic Expressions
Eric Weisstein's MathWorld, Clausen's Integral.
Eric Weisstein's MathWorld, Polylogarithm.

Cf. A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; Pi^2/8*Log[2] - 7/48*Zeta[3] + 11/24*Pi*Clausen2[Pi/6] - 29/24*Pi*Clausen2[5*Pi/6] // RealDigits[#, 10, 101]& // First

Cl2(x)=imag(polylog(2,exp(x*I)));
Pi^2/8*log(2) - 7/48*zeta(3) + 11/24*Pi*Cl2(Pi/6) - 29/24*Pi*Cl2(5*Pi/6) \\ Charles R Greathouse IV, Aug 27 2014

Pi^2/8*log(2) - 7/48*zeta(3) + 11/24*Pi*Cl2(Pi/6) - 29/24*Pi*Cl2(5*Pi/6), where Cl2 is the Clausen function Cl2(t) = Sum_{n>0} sin(n*t)/n^2.

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

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

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

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A261025 Decimal expansion of Cl_2(Pi/4), where Cl_2 is the Clausen function of order 2.

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

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A244843 Decimal expansion of the integral of log(2+x^2+y^2)/((1+x^2)*(1+y^2)) dx dy over the square [0,1]x[0,1].

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

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