Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 083, 11 pages      arXiv:1509.00886
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Certain Integrals Arising from Ramanujan's Notebooks

Bruce C. Berndt a and Armin Straub b
a) University of Illinois at Urbana-Champaign, 1409 W Green St, Urbana, IL 61801, USA
b) University of South Alabama, 411 University Blvd N, Mobile, AL 36688, USA

Received September 05, 2015, in final form October 11, 2015; Published online October 14, 2015

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.

Key words: Ramanujan's notebooks; contour integration; trigonometric integrals.

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  1. Berndt B.C., Ramanujan's notebooks. Part I, Springer-Verlag, New York, 1985.
  2. Berndt B.C., Ramanujan's notebooks. Part IV, Springer-Verlag, New York, 1994.
  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, reprinted by Chelsea, New York, 1962, reprinted by Amer. Math. Soc., Providence, RI, 2000.
  5. Ramanujan S., Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957.

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