A173623 -id:A173623 - cp's OEIS frontend

A086054 Decimal expansion of Pi*log(2).

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

Offset: 1

Eric W. Weisstein, Jul 07 2003

Madelung constant b2(2), negated.

			2.1775860903036021305006888982376139...

G. Boros and V. H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004 (equation 13.6.6).

Eric Weisstein's World of Mathematics, Madelung Constants

Cf. A000796 (Pi), A002162 (log(2)), A173623.

RealDigits[Pi Log[2],10,120][[1]] (* Harvey P. Dale, Dec 31 2011 *)

Pi*log(2) = -(8/3)*int(log(x)/sqrt(1+4*x-4*x^2), x=0..1). - John M. Campbell, Feb 07 2012
Pi*log(2) = int((x/sin(x))^2, x=0..Pi/2) = int(log(x^2+1)/(x^2+1), x=0..infinity) = int(-log(cos(x)), x=-Pi/2..Pi/2) = int(arctan(1/x)^2, x=0..infinity). - Jean-François Alcover, May 30 2013
From Amiram Eldar, Jul 11 2020: (Start)
Equals Integral_{x=-1..1} arcsin(x) dx / x.
Equals Integral_{x=-Pi/2..Pi/2} x*cot(x) dx. (End)
Equals Integral_{x = 0..oo} log(x^2 + 4)/(x^2 + 4) dx. - Peter Bala, Jul 22 2022
Equals -Im(Polylog(2, 2)). - Mohammed Yaseen, Jul 03 2024

Corrected by Antti Ahti (antti.ahti(AT)tkk.fi), Nov 17 2004

More terms from Benoit Cloitre, May 21 2005

A196878 Decimal expansion of (Pi/8)(6zeta(3)+Pi^2log(2)+4log(2)^3).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Oct 07 2011

The absolute value of the integral {x=0..Pi/2} log(sin(x))^3 dx. The absolute value of m=3 of sqrt(Pi)/2*(d^m/da^m(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - Seiichi Kirikami and Peter J. C. Moses, Oct 07 2011

			6.041882909775093522150424130675995...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1

G. C. Greubel, Table of n, a(n) for n = 1..5000
K. S. Kolbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{3,0}

Cf. A173623, A196877.

Pi/8*(6*Zeta(3)+Pi^2*log(2)+4*log(2)^3) ; evalf(%) ; # R. J. Mathar, Oct 08 2011

RealDigits[N[Pi/8 (6 Zeta[3] + Pi^2 Log[2] + 4 Log[2]^3), 150]][[1]]
Sqrt[Pi]/2*Derivative[3][Gamma[(#+1)/2]/Gamma[#/2+1]&][0] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 25 2013 *)

Pi/8*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3) \\ G. C. Greubel, Feb 12 2017

Equals A019675*(6*A002117 + A002388*A002162 + 4*A002162^3).

A193716 Decimal expansion of Pi^3log(2)/24 - 3Pi*zeta(3)/16.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Aug 03 2011

The absolute value of the integral {x=0..Pi/2} x^2*log(sin(x )) dx or (d^2/da^2 (integral {x=0..Pi/2} cos(ax)*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^2/da^2 (sin((a+2n)*Pi/2)/n/(a+2n)))))-(Pi/2)^3*log(2)/3. [Seiichi Kirikami and Peter J. C. Moses]

			0.18742642282823108026...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 1.441.2, 4th edition, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).

R. E. Crandall, J. P. Buhler, On the evaluation of Euler sums, Exper. Math. 3 (4) (1994) 275 (discuss int_{0..1} x^n*cot(x) dx which is obtained by partial integration).
S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.

Cf. A173623, A173624, A193717.

RealDigits[ N[Pi (2 Pi^2 Log[2] - 9 Zeta[3]) / 48, 105] ][[1]]

Pi^3*log(2)/24 - 3*Pi*zeta(3)/16 \\ Michel Marcus, Oct 25 2017

Equals A091925*A002162/24-3*A000796*A002117/16.

A193717 Decimal expansion of Pi^4log(2)/64 - 9Pi^2zeta(3)/64 + 93zeta(5)/128.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Aug 03 2011

The absolute value of the integral {x=0..Pi/2} x^3*log(sin(x )) dx or (d^3/da^3 (integral {x=0..Pi/2} sin(ax )*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^3/da^3 ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(Pi/2)^4*log(2)/4. [Seiichi Kirikami and Peter J. C. Moses]

			-0.14002410170685231710...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 4th edition, 1.441.2, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).

S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.

Cf. A173623, A173624, A193716.

RealDigits[N[(2 Pi^4 Log[2] - 18 Pi^2 Zeta[3] + 93 Zeta[5]) / 128, 105]][[1]]

Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128 \\ Michel Marcus, Oct 25 2017

Equals A092425*A002162/64-9*A002388*A002117/64+93*A013663/128.

A196877 Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Oct 07 2011

The value of the integral_{x=0..Pi/2} log(sin(x))^2 dx. The value of sqrt(Pi)/2*(d^2/da^2(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - Seiichi Kirikami and Peter J. C. Moses, Oct 07 2011

			2.04662202447274064616964100817...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1

G. C. Greubel, Table of n, a(n) for n = 1..10000
K. S. Kolbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{2,0}

Cf. A002162, A019669, A072691, A173623, A196878.

RealDigits[N[Pi/2 (Pi^2/12 + Log[2]^2),150]][[1]]

Pi/2*(Pi^2/12+(log(2))^2) \\ Michel Marcus, Jan 13 2015

Equals A019669*(A072691 + A002162^2).

Equals Integral_{x=0..1} log(x)^2/sqrt(1-x^2) dx. - Amiram Eldar, May 27 2023

A093753 Decimal expansion of (-2Catalan + Pilog(2))/2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 15 2004

			0.17282745097458205019574...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Su Hu, Min-soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.011247 (2023), Example 2.5.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 215.
Eric Weisstein's World of Mathematics, Radial Integrals.

Cf. A001008, A002805, A006752, A093754, A173623.

evalf(-Catalan+Pi*log(2)/2) ; # R. J. Mathar, Apr 01 2010

First[RealDigits[Pi*Log[2]/2 - Catalan, 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

Pi*log(2)/2 - Catalan \\ Michel Marcus, Sep 22 2014

Equals Integral_{x=0..1; y=0..1} [x^2+y^2>1]/(x^2+y^2) where [] is the Iverson bracket.
Equals Integral_{0..1} log(1+x^2)/(1+x^2) dx. - Jean-François Alcover, Sep 22 2014
Equals Sum_{k>=1} (-1)^(k+1) * H(k)/(2*k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jul 22 2020

A194656 Decimal expansion of (2Pi^5log(2) - 30Pi^3zeta(3) + 225Pizeta(5))/320.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Sep 01 2011

The absolute value of the integral{x=0..Pi/2} x^4*log(sin(x )) dx or(d^4/da^4(integral {x=0..Pi/2} cos(ax)*log(sin(x )) dx)) at a=0. The absolute value of m=2 of (-1)^(m+1)*(sum {n=1..infinity} (limit {a -> 0} (d^(2m)/da^(2m)(sin((a+2n)*Pi/2)/n/(a+2n)))))-(Pi/2)^(2m+1)*log(2)/(2m+1). - Seiichi Kirikami and Peter J. C. Moses, Sep 01 2011

			0.12204729588592872163...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 1.441.2

Cf. A173623, A173624, A193716, A193717.

RealDigits[ N[Pi (2 Pi^4*Log[2]-30 Pi^2*Zeta[3]+225 Zeta[5])/320, 150]][[1]]

Equals (2*A092731*A002162-30*A091925*A002117+225*A000796*A013663)/320.

A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Sep 01 2011

The absolute value of the integral {x=0..Pi/2} x^5*log(sin(x )) dx or (d^5/da^5 (integral {x=0..Pi/2} sin(ax)*log(sin(x )) dx)) at a=0. The absolute value of m=2 of (-1)^(m+1)*(sum {n=1..infinity} (limit {a -> 0} (d^(2m+1)/da^(2m+1) ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(pi/2)^2(m+1)*log(2)/2/(m+1).

			0.11757583407233248206...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 1.441.2

Cf. A173623, A173624, A193716, A193717, A196456.

RealDigits[ N[(4 Pi^6*Log[2]-90 Pi^4*Zeta[3]+1350 Pi^2*Zeta[5]-5715 Pi^2*Zeta[7])/1536,150]][[1]]

Equals (4*A092732*A002162-90*A092425*A002117+1350*A002388*A013663-5715*A013665)/1536.

A374952 Decimal expansion of 7zeta(3)/16 + Pi^2log(2)/8, where zeta is the Riemann zeta function.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 04 2024

			1.38103595311446206796832033990552137987215388392245...

Cf. A173623, A173624.

7*Zeta(3)/16 + Pi^2*log(2)/8 ; evalf(%) ;

RealDigits[7*Zeta[3]/16 + Pi^2*Log[2]/8, 10, 120][[1]] (* Amiram Eldar, Aug 05 2024 *)

Equals the absolute value of Integral_{x=0..Pi/2} x*log(cos x) dx.

Equals (Pi/2) * A173623 - A173624.

A375594 Decimal expansion of Pi(Pi^2log(2) + 4log(2)^3 + 6zeta(3))/48.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 20 2024

Apart from a factor sqrt(Pi)/16 the same as Adamchik's generalized Stirling number [1/2,4].

			1.006980484962515...

V. S. Adamchik, On Stirling numbers and Euler sums, J. Comput. Appl. Math. 79 (1) (1997) 119-130.
R. J. Mathar, Chebyshev approximation of x^m*(-log x)^l in the interval 0<=x<=1, arXiv:2408.15212 [math.CA] (2024).

Cf. A019669 (2F1), A173623 (3F2), A318741 (4F3).

1/48*Pi*(Pi^2*log(2)+4*log(2)^3+6*Zeta(3)) ; evalf(%) ;

First[RealDigits[Pi*(Pi^2*Log[2] + 4*Log[2]^3 + 6*Zeta[3])/48, 10, 100]] (* Paolo Xausa, Aug 23 2024 *)

Equals 5F4(1/2,1/2,1/2,1/2,1/2; 3/2,3/2,3/2,3/2; 1) = Sum_{k>= 0} binomial(2k,k)/[2^(2k)*(2k+1)^4].

Equals A196878/6. - R. J. Mathar, Aug 23 2024

cp's OEIS Frontend

A086054 Decimal expansion of Pi*log(2).

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A196878 Decimal expansion of (Pi/8)*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3).

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A193716 Decimal expansion of Pi^3*log(2)/24 - 3*Pi*zeta(3)/16.

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A193717 Decimal expansion of Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128.

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A196877 Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).

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A093753 Decimal expansion of (-2*Catalan + Pi*log(2))/2.

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A196878 Decimal expansion of (Pi/8)(6zeta(3)+Pi^2log(2)+4log(2)^3).

A193716 Decimal expansion of Pi^3log(2)/24 - 3Pi*zeta(3)/16.

A193717 Decimal expansion of Pi^4log(2)/64 - 9Pi^2zeta(3)/64 + 93zeta(5)/128.

A093753 Decimal expansion of (-2Catalan + Pilog(2))/2.

A194656 Decimal expansion of (2Pi^5log(2) - 30Pi^3zeta(3) + 225Pizeta(5))/320.

A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.

A374952 Decimal expansion of 7zeta(3)/16 + Pi^2log(2)/8, where zeta is the Riemann zeta function.

A375594 Decimal expansion of Pi(Pi^2log(2) + 4log(2)^3 + 6zeta(3))/48.