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{{italic title}}
{{Infobox book series
| name = Princeton Lectures in Analysis
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== History ==
The first author, [[Elias M. Stein]], is a [[mathematician]] who has made significant research contributions to the field of [[mathematical analysis]].
Beginning in the spring of 2000, Stein taught a sequence of four intensive undergraduate courses in analysis at [[Princeton University]], where he was a 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 2000 the first manuscript was nearly finished. That fall Stein taught the course in [[complex analysis]] while he and Shakarchi worked on the corresponding manuscript. Paul Hagelstein, then a [[postdoctoral scholar]] in the Princeton math department, was a teaching assistant for this course. In spring 2001, 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.<ref name=fefferman>{{cite article |first1=Charles |last1=Fefferman |authorlink1=Charles Fefferman |first2=Robert |last2=Fefferman |authorlink2=Robert Fefferman |first3=Paul |last3=Hagelstein |first4=Nataša |last4=Pavlović |first5=Lillian |last5=Pierce |title=Princeton Lectures in Analysis by Elias M. Stein and Rami Shakarchi—a book review |journal=Notices of the [[American Mathematical Society|AMS]] |volume=59 |number=5 |month=May |year=2012 |pages=641–47 |url=http://www.ams.org/notices/201205/rtx120500641p.pdf |accessdate=Sep. 16, 2014}}</ref> The project received financial support from Princeton University and from the [[National Science Foundation]].<ref>Page ix of all four Stein & Shakarchi volumes.</ref>
Shakarchi earned his Ph.D. from Princeton in 2002<ref name=duren>{{cite article |first=Peter |last=Duran |authorlink=Peter Duren |title=Princeton Lectures in Analysis. By Elias M. Stein and Rami Shakarchi |journal=[[American Mathematical Monthly]] |volume=115 |number=9 |month=Nov |year=2008 |pages=863–66}}</ref> and moved to [[London]] to work in finance. Nonetheless he continued working on the books, even as his employer, [[Lehman Brothers]], [[Bankruptcy of Lehman Brothers|collapsed]] in 2008.<ref name=fefferman/> The first two volumes were published in 2003. The third followed in 2005, and the fourth in 2011.
== Contents ==
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. They also provide applications of the theory to other fields of mathematics, particularly [[partial differential equation]]s and [[number theory]].<ref name=fefferman/><ref name=duren/>
''Fourier Analysis'' covers the [[Discrete Fourier transform|discrete]], [[Continuous Fourier transform|continuous]], and [[Finite Fourier transform|finite]] [[Fourier transform]]s and their properties, including inversion. It also presents applications to partial differential equations, [[Dirichlet's theorem on arithmetic progressions]], and other topics.<ref>Stein & Shakarchi, ''Fourier Analysis''.</ref> Because [[Lebesgue integration]] is not introduced until the third book, the authors use [[Riemann integration]] in this volume.<ref name=duren/> They begin with Fourier analysis because of its central role within the historical development and contemporary practice of analysis.<ref name=gouvea/>
''Complex Analysis'' treats the standard topics of a course in complex variables as well as several applications to other areas of mathematics.<ref name=fefferman/><ref name=shiu/> The chapters cover the [[complex plane]], [[Cauchy's integral theorem]], [[meromorphic function]]s, connections to Fourier analysis, [[entire function]]s, the [[gamma function]], the [[Riemann zeta function]], [[conformal map]]s, [[elliptic function]]s, and [[theta function]]s.<ref>Stein & Shakarchi, ''Complex Analysis''.</ref>
''Real Analysis'' begins with [[measure theory]], Lebesgue integration, and [[differentiation]] in [[Euclidean space]]. It then covers [[Hilbert space]]s before returning to measure and integration in the context of abstract measure spaces. It concludes with a chapter on [[Hausdorff measure]] and [[fractal]]s.<ref>Stein & Shakarchi, ''Real Analysis''.</ref>
''Functional Analysis'' has chapters on several advanced topics in analysis: [[Lp space|L<sup>''p''</sup> spaces]], [[Distribution (mathematics)|distributions]], the [[Baire category theorem]], [[probability theory]] including [[Brownian motion]], [[several complex variables]], and [[oscillatory integral]]s.<ref>Stein & Shakarchi, ''Functional Analysis''.</ref>
== Reception ==
The books "received rave reviews indicating they are all outstanding works written with remarkable clarity and care."<ref name=oconnor/>
[[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. He mentioned in particular geometric aspects of complex analysis covered in [[Lars Ahlfors]]'s textbook but noted that Stein and Shakarchi also treat some topics Ahlfors skips.<ref name=duren/>
== List of books ==
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