A173623 -id:A173623

A086054

Decimal expansion of Pi*log(2).

+10
9

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Madelung constant b2(2), negated.

REFERENCES

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).

LINKS

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Madelung Constants

FORMULA

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

EXAMPLE

2.1775860903036021305006888982376139...

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,cons,easy

AUTHOR

Eric W. Weisstein, Jul 07 2003

EXTENSIONS

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

More terms from Benoit Cloitre, May 21 2005

STATUS

approved

A196878

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

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

REFERENCES

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

LINKS

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}

FORMULA

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

EXAMPLE

6.041882909775093522150424130675995...

MAPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A173623, A196877.

KEYWORD

cons,nonn

AUTHOR

Seiichi Kirikami, Oct 07 2011

STATUS

approved

A193716

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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]

REFERENCES

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).

LINKS

Table of n, a(n) for n=0..99.

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.

FORMULA

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

EXAMPLE

0.18742642282823108026...

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A173623, A173624, A193717.

KEYWORD

cons,nonn

AUTHOR

Seiichi Kirikami, Aug 03 2011

STATUS

approved

A193717

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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]

REFERENCES

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).

LINKS

Table of n, a(n) for n=0..99.

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

FORMULA

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

EXAMPLE

-0.14002410170685231710...

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A173623, A173624, A193716.

KEYWORD

cons,nonn

AUTHOR

Seiichi Kirikami, Aug 03 2011

STATUS

approved

A196877

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

REFERENCES

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

LINKS

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}

FORMULA

Equals A019669*(A072691 + A002162^2).

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

EXAMPLE

2.04662202447274064616964100817...

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

cons,nonn

AUTHOR

Seiichi Kirikami, Oct 07 2011

STATUS

approved

A093753

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

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..101.

Su Hu, Min-soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.011247 (2023), Example 2.5.

Eric Weisstein's World of Mathematics, Radial Integrals

FORMULA

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

EXAMPLE

0.17282745097458205019574...

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Apr 15 2004

STATUS

approved

A194656

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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]

REFERENCES

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

LINKS

Table of n, a(n) for n=0..104.

FORMULA

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

EXAMPLE

0.12204729588592872163...

MATHEMATICA

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

CROSSREFS

Cf. A173623, A173624, A193716, A193717.

KEYWORD

cons,nonn

AUTHOR

Seiichi Kirikami, Sep 01 2011

STATUS

approved

A194657

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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).

REFERENCES

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

LINKS

Table of n, a(n) for n=0..104.

FORMULA

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

EXAMPLE

0.11757583407233248206...

MATHEMATICA

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]]

CROSSREFS

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

KEYWORD

cons,nonn

AUTHOR

Seiichi Kirikami, Sep 01 2011

STATUS

approved

A374952

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..103.

FORMULA

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

Equals (Pi/2) * A173623 - A173624.

EXAMPLE

1.38103595311446206796832033990552137987215388392245...

MAPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A173623, A173624.

KEYWORD

nonn,cons

AUTHOR

R. J. Mathar, Aug 04 2024

STATUS

approved

A375594

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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

LINKS

Table of n, a(n) for n=1..102.

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 (2024)

FORMULA

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

EXAMPLE

1.006980484962515...

MAPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

R. J. Mathar, Aug 20 2024

STATUS

approved