Ramanujan's Beautiful Integrals

Bruce C Berndt (1) , Atul Dixit (2)

1 Department of Mathematics, Illinois State University
2 IIT Gandhinagar - Indian Institute of Technology [Gandhinagar]

Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the Journal of the Indian Mathematical Society, notebooks, Quarterly Reports to the University of Madras, letters to Hardy, published papers and the Lost Notebook. His evaluations are often surprising, beautiful, elegant, and useful in other mathematical contexts. He also discovered general methods for evaluating and approximating integrals. A survey of Ramanujan's contributions to the evaluation of integrals is given, with examples provided from each of the above-mentioned sources.

Mots clés: Elliptic integrals; Frullani's Theorem; Riemann zeta function; beta and q-beta integrals; lemniscate integral; theta functions; modular equations

Domaines: Mathématiques [math]

Bruce C Berndt, Atul Dixit. Ramanujan's Beautiful Integrals. Hardy-Ramanujan Journal, 2021, Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2020, pp.69-82. ⟨10.46298/hrj.2021.7429⟩. ⟨hal-03208369⟩