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Publisher:

Springer Verlag

Publication Date:

2005

Number of Pages:

437

Format:

Hardcover

Price:

89.95

ISBN:

0-387-25529-X

Category:

Monograph

[Reviewed by , on ]

Donald L. Vestal

09/10/2006

Those familiar with the works of Indian genius Srinivasa Ramanujan will recognize Bruce Berndt as the author of the five-volume work *Ramanujan's Notebooks*. In that collection, he combed through the notebooks of Ramanujan, presenting his results and findings, and providing proofs for them. In *Ramanujan’s Lost Notebook* , Berndt is joined by Andrews in the first of (what will be “approximately”) four volumes dedicated to the “lost notebook” of Ramanujan.

The “lost notebook” was in fact a 138-page manuscript found in materials from the estate of G. N. Watson. The manuscript, written in “Ramanujan’s distinctive handwriting,” contained over 600 formulas (without proof). The authors have taken these results, provided proofs, placed them in the context of contemporary mathematics, and organized them accordingly.

Most of this volume deals with *q*-series. The topics include the Rogers-Ramanujan continued fraction, other continued fractions, the Rogers-Fine Identity, some Rogers-Ramanujan-Slater-Type Identities, partial fractions, the use of Hadamard products for *q*-series, integrals of theta functions, evaluation of some elliptic integrals, and modular equations developed from the eta function. Although little of the material is new, it is well organized and extremely well documented with just over 300 references.

This book is not for the faint of heart—it is certainly not “light reading.” Here is a small taste of the kind of material to be found in this book. This book is for the true fans of series or scholars of Ramanujan (Ramanuphiles?). If you enjoyed the original *Ramanujan’s Notebook* series, then it’s hard to pass this up.

Donald L. Vestal is Assistant Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)sdstate.edu

Preface.- Introduction.- The Rogers-Ramanujan Continued Fraction and Its Modular Properties.- Explicit Evaluations of the Rogers-Ramanujan Continued Fraction.- A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions.- The Rogers-Ramanujan Continued Fraction and Its Connections with Partitions and Lambert Series.- Finite Rogers-Ramanujan Continued Fractions.- Other q-continued Fractions.- Asymptotic Formulas for Continued Fractions.- Ramanujan's Continued Fraction for (q2; q3)8/(q; q3)8.- The Rogers-Fine Identity.- An Empirical Study of the Rogers-Ramanujan Identities.- Rogers-Ramanujan-Slater Type Identities.- Partial Fractions.- Hadamard Products for Two q-Series.- Integrals of Theta-functions.- Incomplete Elliptic Integrals.- Infinite Integrals of q-Products.- Modular Equations in Ramanujan's Lost Notebook.- Fragments on Lambert Series.- Location Guide.- Provenance.- References.- Index.

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