Certain Integrals Arising from Ramanujan's Notebooks
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015)
Voir la notice de l'article provenant de la source Math-Net.Ru
In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \mathrm{d} x + \frac{\pi}{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d}x = 0. $$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by $\log^2x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples.
Keywords:
Ramanujan's notebooks; contour integration; trigonometric integrals.
Bruce C. Berndt; Armin Straub. Certain Integrals Arising from Ramanujan's Notebooks. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a82/
@article{SIGMA_2015_11_a82,
author = {Bruce C. Berndt and Armin Straub},
title = {Certain {Integrals} {Arising} from {Ramanujan's} {Notebooks}},
journal = {Symmetry, integrability and geometry: methods and applications},
year = {2015},
volume = {11},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a82/}
}
[1] Berndt B. C., Ramanujan's notebooks, v. I, Springer-Verlag, New York, 1985 | DOI | MR | Zbl
[2] Berndt B. C., Ramanujan's notebooks, v. IV, Springer-Verlag, New York, 1994 | DOI | MR | Zbl
[3] Gradshteyn I. S., Ryzhik I. M., Table of integrals, series, and products, 8th ed., Academic Press Inc., San Diego, CA, 2014
[4] Ramanujan S., Collected papers, Cambridge University Press, Cambridge, 1927 ; Chelsea, New York, 1962; Amer. Math. Soc., Providence, RI, 2000 | Zbl
[5] Ramanujan S., Notebooks, v. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957 | MR | Zbl