On incomplete elliptic integrals related to Ramanujan's theta functions
In his lost notebook Ramanujan mentioned on pages 51-53 certain incomplete elliptic integral identities of first kind. By using theory of modular forms, \textit{S. Raghavan} and \textit{S. S. Rangachari} [in: Number theory and related topics. Papers presented at the Ramanujan birth centenary international colloquium, Bombay, India, 4-11 January, 1988. Oxford: Oxford University Press, published for the Tata Institute of Fundamental Research. 119--149 (1989; Zbl 0748.33013)] had proved Ramanujan's elliptic integral identities. These identities were also proved by \textit{B. C. Berndt} et al. [Contemp. Math. 254, 79--126 (2000; Zbl 0971.33012)] by using another methods. It has been observed a common fact in both proofs that it is setting up some differential equations satisfied by the theta function in consideration. Using this, the authors establish and prove three new incomplete elliptic integrals of first kind.
scientific article; zbMATH DE number 7608550
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