Lectures on Complex Integration

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Part I: Complex Variables. Subscribe to this collection. Need help getting started? The Princeton Lectures in Analysis has been identified as a well written and influential series of textbooks, suitable for advanced undergraduates and beginning graduate students in mathematics. The first author, Elias M.

Lectures on Complex Integration

Stein , was a mathematician who made significant research contributions to the field of mathematical analysis. Before he had authored or co-authored several influential advanced textbooks on analysis. Beginning in the spring of , Stein taught a sequence of four intensive undergraduate courses in analysis at Princeton University , where he was a mathematics professor.

At the same time he collaborated with Rami Shakarchi, then a graduate student in Princeton's math department studying under Charles Fefferman , to turn each of the courses into a textbook.

Stein taught Fourier analysis in that first semester, and by the fall of the first manuscript was nearly finished. That fall Stein taught the course in complex analysis while he and Shakarchi worked on the corresponding manuscript.

Lectures on Complex Integration : A.O. Gogolin :

Paul Hagelstein, then a postdoctoral scholar in the Princeton math department, was a teaching assistant for this course. In spring , when Stein moved on to the real analysis course, Hagelstein started the sequence anew, beginning with the Fourier analysis course. Hagelstein and his students used Stein and Shakarchi's drafts as texts, and they made suggestions to the authors as they prepared the manuscripts for publication.

Shakarchi earned his Ph. Nonetheless he continued working on the books, even as his employer, Lehman Brothers , collapsed in The third followed in , and the fourth in Princeton University Press published all four. The volumes are split into seven to ten chapters each. Each chapter begins with an epigraph providing context for the material and ends with a list of challenges for the reader, split into Exercises, which range in difficulty, and more difficult Problems.

Throughout the authors emphasize the unity among the branches of analysis, often referencing one branch within another branch's book.

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They also provide applications of the theory to other fields of mathematics, particularly partial differential equations and number theory. Fourier Analysis covers the discrete , continuous , and finite Fourier transforms and their properties, including inversion. It also presents applications to partial differential equations, Dirichlet's theorem on arithmetic progressions , and other topics.

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Complex Analysis treats the standard topics of a course in complex variables as well as several applications to other areas of mathematics. Real Analysis begins with measure theory , Lebesgue integration, and differentiation in Euclidean space.

infitipor.gq It then covers Hilbert spaces before returning to measure and integration in the context of abstract measure spaces. It concludes with a chapter on Hausdorff measure and fractals. Functional Analysis has chapters on several advanced topics in analysis: L p spaces , distributions , the Baire category theorem , probability theory including Brownian motion , several complex variables , and oscillatory integrals.

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The books "received rave reviews indicating they are all outstanding works written with remarkable clarity and care. Peter Duren compared Stein and Shakarchi's attempt at a unified treatment favorably with Walter Rudin 's textbook Real and Complex Analysis , which Duren calls too terse. On the other hand, Duren noted that this sometimes comes at the expense of topics that reside naturally within only one branch.