A086054 -id:A086054 - cp's OEIS frontend

A173623 Decimal expansion of Pi*log(2)/2.

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

Offset: 1

R. J. Mathar, Nov 08 2010

			1.08879304515180106525034444...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.3, p. 22.
Paul J. Nahin, Inside Interesting Integrals, Springer 2015, ISBN 978-1493912766.

Su Hu and Min-Soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.01124 [math.NT], 2022.
K. S. Kölbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{1,0}.
Richard J. Mathar, Chebyshev approximation of x^m(-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.
Kazuhiro Onodera, Generalized log sine integrals and the Mordell-Tornheim zeta values, Trans. Am. Math. Soc. 363 (3) (2010), 1463-1485.

Cf. A000984, A086054, A164583.

Maple
```
Pi/2*log(2) ; evalf(%) ;
```

RealDigits[Pi*Log[2]/2, 10, 100][[1]] (* Amiram Eldar, Jul 13 2020 *)

Pi*log(2)/2 \\ Stefano Spezia, Oct 21 2024

Equals abs(Integral_{x=0..Pi/2} log(sin(x)) dx).
Equals A086054 / 2.
From Amiram Eldar, Jul 13 2020: (Start)
Equals Sum_{k>=0} binomial(2*k,k)/(4^k*(2*k+1)^2) = Sum_{k>=0} A000984(k)/A164583(k).
Equals Integral_{x=0..1} arcsin(x)/x dx.
Equals Integral_{x=0..Pi/2} x*cot(x) dx. (End)
Equals Integral_{x = 0..1} log(x + 1/x)/(1 + x^2) dx (Nahin, 2.4.4) = (1/2)*Integral_{x = 0..oo} log(x^2 + 4)/(x^2 + 4) dx = (1/2)*Integral_{x = 0..oo} log(x^2 + 1)/(x^2 + 1) dx = Integral_{x = 0..oo} log(x^2 + 64)/(x^2 + 64) dx. - Peter Bala, Jul 22 2022
Equals 3F2(1/2,1/2,1/2 ; 3/2,3/2 ; 1). - R. J. Mathar, Aug 19 2024

A102886 Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 15 2005

Named after the French mathematician Joseph-Alfred Serret (1819-1885). - Amiram Eldar, May 30 2021

			0.27219826128795026631258611227970174341732296254616...

Eric Billault et al, MPSI- Khôlles de Maths, Ellipses, 2012, exercice 11.10, pp. 252-264.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (94) on page 18.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 4.291.8.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (2.2.4)
J.-A. Serret, Note sur l'intégrale Integral_{x=0..1} log(x+1)/(x^2+1) dx, Journal de Mathématiques Pures et Appliquées, Vol. 9 (1844), page 436.
Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 308-309.
Eric Weisstein's World of Mathematics, Serret's Integral.

Cf. A086054 (Pi*log(2)).

RealDigits[Pi*Log[2]/8, 10, 102][[1]] (* Jean-François Alcover, May 17 2013 *)

Pi*log(2)/8 \\ Michel Marcus, Apr 23 2020

intnum(x=0, 1, log(x+1)/(x^2+1)) \\ Michel Marcus, Apr 26 2020

Equals Integral_{x=0..1} arctan(x)/(x+1) dx. - Jean-François Alcover, Mar 25 2013
Equals Integral_{x=0..Pi/4} log(tan(x)+1) dx [see link J.-A. Serret and reference Billault]. - Bernard Schott, Apr 23 2020
Equals Pi*log(2)/8 = Sum_{n>0} (-1)^(n+1) * H(2n) / (2n+1) = H(2)/3 - H(4)/5 + H(6)/7 -... with H(n) = Sum_{j=1..n} 1/j the harmonic numbers. [Jolley]; improved by Bernard Schott, Apr 24 2020
Equals -Integral_{x=0..1} x*arccos(x)*log(x) dx. - Amiram Eldar, May 30 2021
Equals Integral_{x=0..log(2)} x/(e^x + 2*e^(-x) - 2) dx = -Integral_{x=0..Pi/2} log(sin(x))*sin(x)/sqrt(1+sin(x)^2) dx = Integral_{x=0..1} log((1 - x)/x)/(1 + x^2) dx = Integral_{x=0..Pi/4} x/((cos(x) + sin(x))*cos(x)) dx = Integral_{x=0..Pi/4} log(cot(x) - 1) dx (see Shamos). - Stefano Spezia, Nov 13 2024

A217459 Decimal expansion of 2^Pi.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Oct 03 2012

			8.8249778270762876238564296042080015817044108152714849266....

"SackVideo", What is 2^π?, 2022 video.

Cf. A000796, A086054.

RealDigits[2^Pi, 10, 100][[1]] (* Amiram Eldar, Nov 24 2020 *)

fpprec : 100; ev(bfloat(2^(%pi)));  /* Martin Ettl, Oct 04 2012 */

default(realprecision,2000);2^Pi \\ Anders Hellström, Nov 11 2015

Sage
```
2^(pi).n(digits=100)
```

Equals exp(A086054). - Amiram Eldar, Nov 24 2020

A352769 Decimal expansion of Pi^2 * log(2).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 02 2022

Rainer and Serene (1976) used the sum that is given in the first formula in the calculation of the free energy of superfluid Helium-3. They evaluated the sum by 6.8.
Rainwater (1978) found the integral representation of this sum, which is given in the second formula, and evaluated it by 6.84109 +- 0.00001.
Glasser and Ruehr (1981) proved the sum is equal to this constant.

			6.84108846385711654484747915395409607129977904818791...

Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, Philadelphia, 1987, pp. 187-188.

D. Bierens de Haan, Nouvelles tables d'intégrales définies, Leide, 1867, Table 206, eq. 10.
M. L. Glasser, An Infinite Triple Summation, Problem 80-13, SIAM Review, Vol. 22, No. 3 (1980), pp. 363-364; Solutions by the proposer and by O. G. Ruehr, ibid., Vol. 23, No. 3 (1981), pp. 393-394.
C. F. Lindman, Examen des nouvelles tables d'intégrales définies de m. Bierens de Haan, Amsterdam 1867, Stockholm, 1891, p. 99, Tab. 206, 10.
D. Rainer and J. W. Serene, Free energy of superfluid He 3, Phys. Rev. B, Vol. 13, No. 11 (1976), pp. 4745-4765; Erratum, ibid., Vol. 18, No. 7 (1978), p. 3760.
J. C. Rainwater, Evaluation of frequency sums for the free energy of superfluid He 3, Phys. Rev. B, Vol. 18, No. 7 (1978), pp. 3728-3729.

Cf. A000796 (Pi), A002162 (log(2)), A002117 (zeta(3)), A002388 (Pi^2), A086054 (Pi*log(2)).

Mathematica
```
RealDigits[Pi^2*Log[2], 10, 100][[1]]
```

Pi^2 * log(2) \\ Michel Marcus, Apr 02 2022

Equals Sum_{i,j,k, positive and negative odd integers} sign(i) * sign(j) * sign(k) * sign(i+j-k)/(i^2*j^2).
Equals -8 * Integral_{x=0..1} arctanh(x)*log(x)/(x*(1-x^2)) dx - 7*zeta(3)/2.
Equals Integral_{x=0..Pi/2} (4*x^2*cos(x) - x*(Pi-x))/sin(x) dx (Bierens de Haan, 1867; Lindman, 1891).

A188141 Decimal expansion of integral ((arctan(1/x))^3,x=0..infinity).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Mar 23 2011

The computation of this integral was mentioned as a challenge by Robert Israel on the newsgroup sci.math (Dec 22 2010), a closed form solution being given by Valeri Astanoff.

			1.9754169..

The Math Forum at Drexel - Valeri Astanoff Re: Nice Integral, Dec 22, 2010.

Cf. A086054 (int(arctan(1/x)^2, x=0..infinity)).

RealDigits[N[(3/8)*(Pi^2*Log[4] - 7*Zeta[3]) , 110]][[1]]
(* or as a numerical check : *)
RealDigits[NIntegrate[ArcTan[1/x]^3, {x, 0, Infinity}, WorkingPrecision -> 110]][[1]] (* Jean-François Alcover, Mar 23 2011 *)
RealDigits[ N[ Integrate[ ArcTan[1/x]^3, {x, 0, Infinity}], 110]][[1]] (* Jean-François Alcover, Oct 19 2012, since version 6.0 *)

A175638 Decimal expansion of the upper limit x such that Integral_{u=0..Pix} ucot(u) du = 0.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

Because the integral from u=0 up to u=Pi/2 equals log(2)*Pi/2 = A086054/2, this is also the x such that Integral_{u=Pi/2..Pi*x} u*cot(u) du = -log(2)*Pi/2. By partial integration, Integral_{u} u*cot(u) du = u*log(sin(u)) - Integral_{u} log(sin(u)) du, used with a Newton method in the Maple implementation.

			x = 0.7912265710...

G. Freiman and H. Halberstam, On a product of sines, Acta Arithmetica 49 issue 4 (1987) 377-385.

intu := proc(u) u*log(sin(u)) - int( log(sin(t)),t=Pi/2..u) ; evalf(%) ; end proc:
Digits := 80 : x := 0.79122 :
for it from 1 to 10 do x0 := intu(evalf(Pi*x))+Pi*log(2)/2 ; xnew := x-evalf(x0)/Pi^2/x/cot(Pi*x) ; x := evalf(xnew) ; print(x) ; end do:

First@ RealDigits@ Re[ FindRoot[ Integrate[ u*Cot[u], {u, 0, x*Pi}], {x, 0.7}, WorkingPrecision -> 2^7][[1, 2]]] (* Robert G. Wilson v, Aug 03 2010 *)

More terms from Robert G. Wilson v, Aug 03 2010

A271872 Decimal expansion of the doubly infinite sum N_3 = Sum_{i,j,k = -inf..inf} (-1)^(i+j+k)/(i^2+j^2+k^2), a lattice constant analog of Madelung's constant (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 24 2016

			-2.51935615208944531334271172732943791211649913675173257750066...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.10 Madelung's constant, p. 77.

Eric Weisstein's MathWorld, Madelung Constants

Cf. A088537 (M_2), A085469 (M_3), A090734 (M_4), A086054 (N_2).

digits = 101; Clear[s]; s[max_] := s[max] = NSum[(-1)^n Csch[Pi *Sqrt[m^2 + 2 n^2]]/Sqrt[m^2 + 2 n^2], {m, 1, max}, {n, 1, max}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 10]; s[10]; s[max = 20]; Print[max]; While[RealDigits[s[max], 10, digits + 5][[1]] != RealDigits[s[max/2], 10, digits + 5][[1]], max = max*2; Print[max]]; N3 = Pi^2/3 - Pi*Log[2] - Pi/Sqrt[2] Log[2 (Sqrt[2] + 1)] + 8 Pi*s[max]; RealDigits[N3, 10, digits][[1]]

N_3 = Pi^2/3-Pi*log(2)-(Pi/sqrt(2))*log(2(sqrt(2)+1))+8 Pi*Sum_{m,n >= 1} (-1)^n csch(Pi*sqrt(m^2+2n^2))/sqrt(m^2+2n^2).

A348563 Decimal expansion of Sum_{k>=1} H(k) * binomial(2k,k)/((2k+1)*4^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 22 2021

			1.48627628640527392971772516149192249575801379675740...

Amrik Singh Nimbran, Sums of series involving central binomial coefficients and harmonic numbers, The Mathematics Student, Vol. 88, No. 1-2 (2019), pp. 125-135; arXiv preprint, arXiv:1806.03998 [math.NT], 2018-2019.

Cf. A000984, A001008, A002805, A006752, A086054.

RealDigits[4*Catalan - Pi*Log[2], 10, 100][[1]]

Equals 4*G - Pi*log(2), where G is Catalan's constant (A006752).
Equals Integral_{-Pi/2 .. Pi/2} log(1+cos(x)) dx. - Philippe Deléham , Jan 12 2024
Equals Integral_{x=0..1} log(1 + sqrt(x))/sqrt(x - x^2) dx. - Kritsada Moomuang, Jun 06 2025

A372919 Decimal expansion of (Pi/4)*log(2) + Catalan.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 16 2024

			1.46036211...

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (5.1.3).
Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 465.

Cf. A006752, A086054.

Maple
```
Pi/4*log(2)+Catalan ; evalf(%) ;
```

RealDigits[Pi*Log[2]/4 + Catalan, 10, 120][[1]] (* Amiram Eldar, May 21 2024 *)

Equals Integral_{x=0..oo} log(x+1)/(x^2+1) dx = A086054/4 + A006752.
From Amiram Eldar, May 21 2024: (Start)
Formulas from Shamos (2011):
Equals Integral_{x>=1} log(x^2-1)/(x^2+1) dx.
Equals Integral_{x>=0} x/(exp(x) + 2*exp(-x) - 2) dx.
Equals Integral_{x=0..Pi/2} (sin(x)-cos(x))/(sin(x)+cos(x)) * x dx.
Equals Integral_{x>=0} arccot(x)/(x+1) dx.
Equals Integral_{x=0..Pi/2} log(1+tan(x)) dx. (End)
Equals Integral_{x=0..oo} arctan(x)/(x*(1 + x)) dx. - Kritsada Moomuang, Jun 18 2025

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A173623 Decimal expansion of Pi*log(2)/2.

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A102886 Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx.

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A217459 Decimal expansion of 2^Pi.

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A352769 Decimal expansion of Pi^2 * log(2).

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A188141 Decimal expansion of integral ((arctan(1/x))^3,x=0..infinity).

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A175638 Decimal expansion of the upper limit x such that Integral_{u=0..Pi*x} u*cot(u) du = 0.

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A271872 Decimal expansion of the doubly infinite sum N_3 = Sum_{i,j,k = -inf..inf} (-1)^(i+j+k)/(i^2+j^2+k^2), a lattice constant analog of Madelung's constant (negated).

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A175638 Decimal expansion of the upper limit x such that Integral_{u=0..Pix} ucot(u) du = 0.

A348563 Decimal expansion of Sum_{k>=1} H(k) * binomial(2k,k)/((2k+1)*4^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.