Geodesic
Certain Integrals Arising from Ramanujan's Notebooks
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@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/}
}
TY  - JOUR
AU  - Bruce C. Berndt
AU  - Armin Straub
TI  - Certain Integrals Arising from Ramanujan's Notebooks
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2015
VL  - 11
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a82/
LA  - en
ID  - SIGMA_2015_11_a82
ER  - 
%0 Journal Article
%A Bruce C. Berndt
%A Armin Straub
%T Certain Integrals Arising from Ramanujan's Notebooks
%J Symmetry, integrability and geometry: methods and applications
%D 2015
%V 11
%U http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a82/
%G en
%F SIGMA_2015_11_a82
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/

[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