Ramanujan Log-Trigonometric Integrals

Author

Eric W. Weisstein
January 20, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/RamanujanLog-TrigonometricIntegrals.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RamanujanLog-TrigonometricIntegrals.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

(SuperMinus[(Subscript[R, n])])

R[n_,-1]:=(2/π)integrate[1/((((θ^2)+((Log[Cos[θ]])^2)))^(2^(-n-1)))Nest[Sqrt[1/2+#/2]&,(Sqrt[(((Log[Cos[θ]])^2)/((θ^2)+((Log[Cos[θ]])^2)))]),n],{θ,0,π/2}]

R[0,-1]/.integrate:>NIntegrate

R[0,-1]/.integrate:>Integrate

1/Log[2.]

R[1,-1]/.integrate:>NIntegrate

1/((Log[2])^(1/2.))

Table[{R[n,-1],((Log[2.])^(-(2^(-n))))},{n,0,5}]/.integrate:>NIntegrate

(SuperPlus[(Subscript[R, n])])

R[n_,1]:=(2/π)integrate[((((θ^2)+((Log[Cos[θ]])^2)))^(2^(-n-1)))Nest[Sqrt[1/2+#/2]&,(Sqrt[(((Log[Cos[θ]])^2)/((θ^2)+((Log[Cos[θ]])^2)))]),n],{θ,0,π/2}]

R[0,1]/.integrate:>NIntegrate

R[0,1]/.integrate:>Integrate

R[1,1]/.integrate:>NIntegrate

FullSimplify[R[1,1],0<θ<π/2]

%42/.integrate:>Integrate

((Log[2])^(1/2.))

Table[{R[n,1],((Log[2.])^(2^(-n)))},{n,0,5}]/.integrate:>NIntegrate