TOPICS
Search

Ramanujan Log-Trigonometric Integrals


Oloa (2010, pers. comm., Jan. 20, 2010) has considered the following integrals containing nested radicals of 1/2 plus terms in theta^2 and ln^2costheta:

R_n^-=2/piint_0^(pi/2)(theta^2+ln^2costheta)^(-2^((-n-1)))sqrt(1/2+1/2sqrt(1/2+...+1/2sqrt((ln^2costheta)/(theta^2+ln^2costheta))))dtheta
(1)
R_n^+=2/piint_0^(pi/2)(theta^2+ln^2costheta)^(2^((-n-1)))sqrt(1/2+1/2sqrt(1/2+...+1/2sqrt((ln^2costheta)/(theta^2+ln^2costheta))))dtheta,
(2)

which he terms Ramanujan log-trigonometric integrals because they involve terms like Ramanujan's nested radicals of 1/2.

The special case

 R_0^+=ln2
(3)

was known to Euler.

Amazingly, the general integrals have closed forms

R_n^-=(ln2)^(-2^(-n))
(4)
R_n^+=(ln2)^(2^(-n))
(5)

for n>=1.


See also

Nested Radical

Explore with Wolfram|Alpha

References

Oloa, O. "Two Conjectures on Ramanujan Log-trigonometric Integrals." Unpublished manuscript. January 2010.

Referenced on Wolfram|Alpha

Ramanujan Log-Trigonometric Integrals

Cite this as:

Weisstein, Eric W. "Ramanujan Log-Trigonometric Integrals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujanLog-TrigonometricIntegrals.html

Subject classifications