## Ramanujan’s Notebooks, Part 3During the time period between 1903 and 1914, Ramanujan worked in almost complete isolation in India. Throughout these years, he recorded his mathematical results without proofs in notebooks. Upon Ramanujan's death in 1920, G.H. Hardy strongly urged that Ramanujan's notebooks be published and edited. The English mathematicians G.N. Watson and B.M. Wilson began this task in 1929, but although they devoted nearly ten years to the project, the work was never completed. In 1957, the Tata Institute of Fundamental Research in Bombay published a photostat edition of the notebooks, but no editing was undertaken. In 1977, Berndt began the tasks of editing Ramanujan's notebooks. Proofs are provided to theorems not yet proven in previous literature, and many results are so startling and different that there are no results akin to them in the literature. |

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### Contents

CHAPTER 20 | 330 |

Modular Equations of Higher and Composite Degrees 325 | 396 |

CHAPTER 21 | 449 |

489 | |

505 | |

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31 of Chapter algebra apply Entry 18(iv Chapter 16 Chapter 20 complete the proof complex numbers continued fraction Corollary cusp deduce defined denote derive desired result elementary calculation elliptic integrals Employing Entries Entries 25(i Entry 19 Entry 22 Entry 31 equation of degree equivalent established expression f(bq formula Fourier series given Hence integer Jacobi triple product Jacobian elliptic functions Lastly left side Lemma MACSYMA modular equations modular forms multiplier system notation obtain OO kq Ordr(F paper poles positive integer proof of Entry PROOF OF vii prove q by q q f(q qp(q quintuple product identity Ramanujan reciprocal Replacing q respectively right side Rogers–Ramanujan identities second equality second notebook Section 31 simplifying Substituting tends theorem theory of modular theta-function identities theta-functions To(n transformation triple product identity Whittaker and Watson