Elementary proof of a formula of Ramanujan
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- by Robert L. Lamphere PDF
- Proc. Amer. Math. Soc. 91 (1984), 416-420 Request permission
Abstract:
In this paper we use only elementary calculus to prove Ramanujan’s integral formula \[ \int _0^\infty {{x^{m - 1}}} \frac {{(1 + abx)(1 + a{b^2}x) \cdots (1 + a{b^n}x) \cdots }}{{(1 + x)(1 + bx)(1 + {b^2}x) \cdots (1 + {b^n}x) \cdots }}dx = \frac {\pi }{{{\operatorname {Sin}}(m\pi )}}\prod \limits _{k = 1}^\infty {\frac {{(1 - {b^{k - m}})(1 - a{b^k})}}{{(1 - {b^k})(1 - a{b^{k - m}})}}} \] where $m$, $a$ and $b$ are positive with $b < 1$ and $a < {b^{m - 1}}$.References
- Richard Askey, Ramanujan’s extensions of the gamma and beta functions, Amer. Math. Monthly 87 (1980), no. 5, 346–359. MR 567718, DOI 10.2307/2321202 Joseph Edwards, Treatise on integral calculus, vol. 2, reprinted Chelsea, New York, 1922. G. H. Hardy, Proof of a formula of Mr. Ramanujan, Messenger of Math. 44 (1915), 18-21. —, Pure mathematics, 1952; reprinted Cambridge University Press, London, 1963. S. Ramanujan, Collected papers, Chelsea, New York, 1962.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 416-420
- MSC: Primary 33A15; Secondary 05A30, 26A42
- DOI: https://doi.org/10.1090/S0002-9939-1984-0744641-1
- MathSciNet review: 744641