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Mathematics - Algebra | Ramanujan's Lost Notebook - Part II

Ramanujan's Lost Notebook

Part II

Volume package: Ramanujan's Lost Notebook

Andrews, George E., Berndt, Bruce C.

2009

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  • The "lost notebook" contains considerable material on mock theta functions and so undoubtedly emanates from the last year of Ramanujan's life
  • It should be emphasized that the material on mock theta functions is perhaps Ramanujan's deepest work
  • Mathematicians are probably several decades away from a complete understanding of those functions

This volume is the second of approximately four volumes that the authors plan to write on Ramanujan’s lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988.

 

The primary topics addressed in the authors’ second volume on the lost notebook are q-series, Eisenstein series, and theta functions. Most of the entries on q-series are located in the heart of the original lost notebook, while the entries on Eisenstein series are either scattered in the lost notebook or are found in letters that Ramanujan wrote to G.H. Hardy from nursing homes.

 

About Ramanujan's Lost Notebook, Volume I:

"Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete."

- Gazette of the Australian Mathematical Society

 

"...the results are organized topically with cross-references to the identities as they appear in the original Ramanujan manuscript. Particularly helpful are the extensive references, indicating where in the literature these results have been proven or independently discovered as well as where and how they have been used."

- Bulletin of the American Mathematical Society

 

"The mathematics community owes a huge debt of gratitude to Andrews and Berndt for undertaking the monumental task of producing a coherent presentation along with complete proofs of the chaotically written mathematical thoughts of Ramanujan during the last year of his life. Some 85 years after his death, beautiful "new" and useful results of Ramanujan continue to be brought to light."

- Mathematical Reviews

Content Level » Research

Keywords » Invariant - approximation - elliptic function - equation - function - identity - theta function - transformation

Related subjects » Algebra - Analysis

Table of contents 

The Heine Transformation.- The Sears– Thomae Transformation.- Bilateral Series.- Well-Poised Series.- Bailey’s Lemma and Theta Expansions.- Partial Theta Functions.- Special Identities.- Theta Function Identities.- Ramanujan’s Cubic Analogue of the Classical Ramanujan–Weber Class Invariants.- Miscellaneous Results on Elliptic Functions and Theta Functions.- Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series.- Two Letters on Eisenstein Series Written from Matlock House.- Eisenstein Series and Modular Equations.- Series Representable in Terms of Eisenstein Series.- Eisenstein Series and Approximations to π.- Miscellaneous Results on Eisenstein Series.

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