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Srinivasa Ramanujan was a mathematician brilliant beyond compare. There is extensive literature available on the work of Ramanujan, but what is more difficult to find in the literature is an analysis that would place his mathematics in context and interpret it in terms of modern developments. The 12 lectures by G. H. Hardy, delivered in 1936, served this purpose at the time they were given. This book presents Ramanujan’s essential mathematical contributions and gives an informal account of some of the major developments that emanated from his work in the 20th and 21st centuries. It contends that his work is still having an impact on many different fields of mathematical research. The book examines some of these themes in the landscape of 21st-century mathematics. These essays, based on the lectures given by the authors, focus on a subset of Ramanujan’s significant papers and show how these papers shaped the course of modern mathematics.  .

Mathematicians -- India -- Biography. --- Ramanujan Aiyangar, Srinivasa, 1887-1920. --- Mathematicians --- Mathematics --- Number theory --- Physical Sciences & Mathematics --- Algebra --- Mathematics - General --- History --- Ramanujan Aiyangar, Srinivasa, --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Mathematics. --- Algebra. --- Fourier analysis. --- Special functions. --- History. --- Number theory. --- Combinatorics. --- Number Theory. --- History of Mathematical Sciences. --- Special Functions. --- Fourier Analysis. --- Scientists --- Functions, special. --- Mathematical analysis --- Analysis, Fourier --- Special functions --- Combinatorics --- Number study --- Numbers, Theory of --- Annals --- Auxiliary sciences of history --- Math --- Science --- India. --- Bharat --- Bhārata --- Government of India --- Ḣindiston Respublikasi --- Inde --- Indi --- Indien --- Indii͡ --- Indland --- Indo --- Republic of India --- Sāthāranarat ʻIndīa --- Yin-tu

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This book discusses major topics in measure theory, Fourier transforms, complex analysis and algebraic topology. It presents material from a mature mathematical perspective. The text is suitable for a two-semester graduate course in analysis and will help students prepare for a research career in mathematics. After a short survey of undergraduate analysis and measure theory, the book highlights the essential theorems that have now become ubiquitous in mathematics. It studies Fourier transforms, derives the inversion theorem and gives diverse applications ranging from probability theory to mathematical physics. It reviews topics in complex analysis and gives a synthetic, rigorous development of the calculus of residues as well as applications to a wide array of problems. It also introduces algebraic topology and shows the symbiosis between algebra and analysis. Indeed, algebraic archetypes were providing foundational support from the start. Multivariable calculus is comprehended in a single glance through the algebra of differential forms. Advanced complex analysis inevitably leads one to the study of Riemann surfaces, and so the final chapter gives the student a hint of these motifs and underlying algebraic patterns.

Algebraic geometry --- Algebraic topology --- Functional analysis --- Harmonic analysis. Fourier analysis --- Differential equations --- Mathematical analysis --- Measuring methods in physics --- Mathematical physics --- Fourieranalyse --- differentiaalvergelijkingen --- analyse (wiskunde) --- topologie (wiskunde) --- complexe veranderlijken --- Fourierreeksen --- mathematische modellen --- meettechniek --- wiskunde

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Number theory --- getallenleer

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This book introduces the reader to the fascinating world of modular forms through a problem-solving approach. As such, besides researchers, the book can be used by the undergraduate and graduate students for self-instruction. The topics covered include q-series, the modular group, the upper half-plane, modular forms of level one and higher level, the Ramanujan τ-function, the Petersson inner product, Hecke operators, Dirichlet series attached to modular forms and further special topics. It can be viewed as a gentle introduction for a deeper study of the subject. Thus, it is ideal for non-experts seeking an entry into the field. .

Mathematics. --- Operator theory. --- Sequences (Mathematics). --- Special functions. --- Number theory. --- Number Theory. --- Operator Theory. --- Special Functions. --- Sequences, Series, Summability. --- Forms, Modular. --- Modular forms --- Forms (Mathematics) --- Functions, special. --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Special functions --- Mathematical analysis --- Functional analysis --- Number study --- Numbers, Theory of

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This book gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to provide a rapid introduction to analytic methods and the ways in which they are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic number theory. The new edition has been completely rewritten, errors have been corrected, and there is a new chapter on equidistribution. About the first edition: "...this monograph gives important results and techniques for specific topics, together with many exercises; it is not possible to describe adequately the wealth of material covered in this book." - Wolfgang Schwarz, Zentralblatt

Number theory --- Discrete mathematics --- discrete wiskunde --- getallenleer