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].

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

0,1

Comments

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.

Links

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.

Formula

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.

Example

0.56959615818361450623645553672717469010787612682122878368281840812485230025...

Mathematica

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

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A019598 A340565 A197283 * A118261 A246749 A357471

Adjacent sequences: A244840 A244841 A244842 * A244844 A244845 A244846

Keyword

cons,nonn

Author

Jean-François Alcover, Jul 07 2014

Status

approved