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Ramanujan’s notebooks. Part III. (English) Zbl 0733.11001
New York etc.: Springer-Verlag. 510 p. DM 148.00 (1991).

Let me begin (as I did in my Zbl review of Part II (1989; Zbl 0716.11001) with an extended excerpt from George Andrews’ review of Part I (1985; Zbl 0555.10001) of this series of books (there will be four in all when Berndt finishes). [The reader is kindly referred to the first two paragraphs of the review of Part II (Editorial note).]

As to the contents, the Introduction contains a fine and careful summary. Then, each chapter has a head note giving a more precise summary and pointing to highlights contained therein.

The second Notebook is an enlarged edition of the first. Part I of the Berndt’s series covered Chapters 1-9 of this second Notebook. In Part II he gave Chapters 10-15. Part III covers Chapters 16-21; this is the end of the organized material in the second Notebook. (In Parts IV and V Berndt will deal with the 100 pages of unorganized material at the end of the second Notebook, 33 similarly unorganized pages of the third Notebook, and all results from the first Notebook note recorded by Ramanujan in the second or third Notebooks.)

“We first briefly review the content of Chapters 16-21. Although theta- functions play the leading role, several other topics make appearances as well. Some of Ramanujan’s most famous theorems are found in Chapter 16. The chapter begins with basic hypergeometric series and some q-continued fractions. In particular, a generalization of the Rogers-Ramanujan continued fraction and a finite version of the Rogers-Ramanujan continued fraction are found. Entry 7 offers an identity from which the Rogers- Ramanujan identities (found in Section 38) can be deduced as limiting cases, a fact that evidently Ramanujan failed to notice. The material on q-series ends with Ramanujan’s celebrated ${}_{1}{\psi }_{1}$ summation. After stating the Jacobi triple product identity, which is a corollary of Ramanujan’s ${}_{1}{\psi }_{1}$ summation, Ramanujan commences his work on theta-functions. Several of this results are classical and well known, but Ramanujan offers many interesting new results, especially in Sections 33-35. ”...

“Chapter 17 begins with Ramanujan’s development of some of the basic theory of elliptic functions highlighted by Entry 6, which provides the basic inversion formula relating theta-functions with elliptic integrals and hypergeometric functions. Section 7 offers many beautiful theorems on elliptic integrals. The following sections are devoted to a catalogue of formulas for the most well-known theta-functions and for Ramanujan’s Eisenstein series, L, M, and N, evaluated at different powers of the argument. These formulas are of central importance in proving modular equations in Chapters 19-21.

Several topics are examined in Chapter 18, although most attention is given to the Jacobian elliptic functions. Approximations to $\pi$ and the perimeter of an ellipse are found. More problems in geometry are discussed in this chapter than in any other chapter. The chapter ends with Ramanujan’s initial findings about modular equations.

Chapters 19 and 20 are devoted to modular equations and associated theta- function identities. Most of the results in these two chapters are new and show Ramanujan at his very best (strongly emphasized by the reviewer). It is here that our proofs undoubtedly often stray from the paths followed by Ramanujan.

Chapter 21 occupies only 4 pages and is the shortest chapter in the second notebook. The content is not unlike that of the previous two chapters, but here the emphasis is on formulas for the series L, M, and N.”

It is not possible in this review to do justice to the care Berndt has given to this work. Suffice it to say that in turning the pages even the casual reader is constantly rewarded with profound and lively editorial remarks concerning two themes: (1) What were the basic insights of Ramanujan that led to these results? (2) What previous (18th and even 17th century) lesser known mathematicians contributed related results to this field (modular forms, functions, equations)? This makes reading fascinating.

The notation used in the book stems largely from q-series; readers not expert in this notation will be required to “translate”; this is facilitated by an Berndt’s excellent cross referencing system. Last, a considerable number of the proofs (Berndt sometimes calls these “verifications” because the identity being proved must be known in advance) rely on the symbolic manipulator MACSYMA. In his first invocation of this tool, Berndt states: “To do this by hand would be a superhuman feat.” (p. 312) And thus the title of the new, widely acclaimed popular biography of Ramanujan: “The Man who knew Infinity”. Indeed, Part III renders the hypothesis that Ramanujan was a mere mortal highly questionable.

##### MSC:
 11-02 Research monographs (number theory) 11-03 Historical (number theory) 11F03 Modular and automorphic functions 33-02 Research monographs (special functions) 14K25 Theta-functions 11F11 Holomorphic modular forms of integral weight 11F20 Dedekind eta function, Dedekind sums 11F27 Theta series; Weil representation; theta correspondences 33Cxx Hypergeometric functions 33Dxx Basic hypergeometric functions
MACSYMA