Princeton Lectures in Analysis: Difference between revisions

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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 [[FiniteDiscrete Fourier transform|finite]]{{disambiguation needed|date=August 2016}} [[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 name=ss1>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 name=ss2>Stein & Shakarchi, ''Complex Analysis''.</ref>