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On an approximate identity of Ramanujan

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Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Zagier, D. (1987). On an approximate identity of Ramanujan. Proceedings of the Indian Academy of Sciences. Mathematical Sciences, 97(1-3), 313-324. doi:10.1007/BF02837833.


Cite as: http://hdl.handle.net/21.11116/0000-0004-395D-5
Abstract
In his second notebook, Ramanujan wrote that 1-qx/(1+q\\sp 2/(1-q\\sp 3x/(1+q\\sp 4/(1-q\\sp 5x/(1+.. = q/(x+q\\sp 4/(x+q\\sp 8/(x+q\\sp12/(x+..\\quad nearly. This paper explains the relationship between these continued fractions and shows that for xgt;0, their difference is asymptotically \\exp (-c(x)/(1-q)) as q→ 1\\sp- where c(x) is monotone decreasing with c(0)=π\\sp 2/4, c(1)=π\\sp 2/5, and in general c(x) can be expressed in terms of the dilogarithm.