Ramanujan's Notebooks, Part IV, by Bruce C. Berndt, is the fourth of five volumes dedicated to editing Ramanujan's notebooks. Ramanujan recorded most of his mathematical discoveries without proofs in notebooks between 1903 and 1914. Although many of his results were already found in the literature, most were not. After Ramanujan's death in 1920, G. N. Watson and B. M. Wilson began to edit his notebooks but never completed the task. A photostat edition was published by the Tata Institute of Fundamental Research in 1957. This volume is the first of two devoted to proving the results found in the unorganized portions of the second and third notebooks, as well as those in the first notebook not found in the second or third. For known results, references are provided; otherwise, complete proofs are given. The book includes chapters on elementary results, number theory, Ramanujan's theory of prime numbers, theta-functions and modular equations, inversion formulas for the lemniscate and allied functions, q-series, integrals, and special functions. It also includes partial fraction expansions, elementary and miscellaneous analysis, and a guide to locating material from the first notebook in the second. The book concludes with references and an index. The work is part of a series aimed at preserving and analyzing Ramanujan's mathematical contributions.Ramanujan's Notebooks, Part IV, by Bruce C. Berndt, is the fourth of five volumes dedicated to editing Ramanujan's notebooks. Ramanujan recorded most of his mathematical discoveries without proofs in notebooks between 1903 and 1914. Although many of his results were already found in the literature, most were not. After Ramanujan's death in 1920, G. N. Watson and B. M. Wilson began to edit his notebooks but never completed the task. A photostat edition was published by the Tata Institute of Fundamental Research in 1957. This volume is the first of two devoted to proving the results found in the unorganized portions of the second and third notebooks, as well as those in the first notebook not found in the second or third. For known results, references are provided; otherwise, complete proofs are given. The book includes chapters on elementary results, number theory, Ramanujan's theory of prime numbers, theta-functions and modular equations, inversion formulas for the lemniscate and allied functions, q-series, integrals, and special functions. It also includes partial fraction expansions, elementary and miscellaneous analysis, and a guide to locating material from the first notebook in the second. The book concludes with references and an index. The work is part of a series aimed at preserving and analyzing Ramanujan's mathematical contributions.