Titlebook: Ramanujan‘s Lost Notebook; Part II George E. Andrews,Bruce C. Berndt Book 2009 Springer-Verlag New York 2009 Invariant.approximation.ellipt

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Partial Theta Functions,from the classical Jacobi theta function ., we have chosen to name the series in (6.1.1) .. We have chosen the designation partial theta functions, in contrast with L.J. Rogers’s “false theta functions” discussed in Chapters 9 and 11 of our first volume [31, pp. 227–239, 256–259].

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Special Identities,The first four identities to be examined have previously been proved [20] by relating them to the theory of Durfee rectangles [13]. We provide an alternative development based on functional equations in Section 7.2.

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,Ramanujan’s Cubic Analogue of the Classical Ramanujan–Weber Class Invariants, elegant values of ., for . ≡ 1 (mod 8). The quantity . can be thought of as an analogue in Ramanujan’s cubic theory of elliptic functions [57, Chapter 33] of the classical Ramanujan–Weber class invariant Gn, which is defined by . where . and . is any positive rational number.

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,Eisenstein Series and Approximations to π, To the right of each integer, Ramanujan recorded a linear equation in .. and ... Although Ramanujan did not indicate the definitions of . and ., we can easily (and correctly) ascertain that . and . are the Eisenstein series . and ., where .. To the right of each equation in .. and .., Ramanujan ent

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iscusses q-series, Eisenstein series, and theta functions.InThis is the second of approximately four volumes that the authors plan to write in their examination of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers. This volume, published by Narosa in 1988, contain

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