Abstract
In his lost notebook, S. Ramanujan recorded incomplete elliptic integral identities of the first kind. In this paper, we give new proofs to Ramanujan’s incomplete elliptic integrals of the first kind for level 5 using the parameter k(q) = R(q)R2(q2), where R(q) is Rogers-Ramanujan continued fraction. Also, we construct new identities for level 5 similar to ones found in Ramanujan’s work. Further, we prove related identities of level 7, using theta function identities.
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Acknowledgement
The author would like to thank Prof. K. R. Vasuki for his advice and guidance during the preparation of this article. Also, the author would like to thank the referee for his valuable comments.
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Bhuvan, E.N. On Ramanujan’s Incomplete Elliptic Integral Identities. Indian J Pure Appl Math 51, 1737–1751 (2020). https://doi.org/10.1007/s13226-020-0493-6
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DOI: https://doi.org/10.1007/s13226-020-0493-6
Key words
- Incomplete elliptic integrals
- modular equations
- Dedekind eta-function
- theta functions
- Rogers-Ramanujan continued fraction