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

Author

Jean-François Alcover, Jul 07 2014

Keywords

• cons
• nonn

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.

Examples

			0.56959615818361450623645553672717469010787612682122878368281840812485230025...
		

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.

Crossrefs

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

Programs

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

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

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.