Complex Analysis (Princeton Lectures in Analysis, No. 2) (Hardcover)

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle.

With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory.

Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences.

The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

隨著第二卷的出版，我們進入了複分析的迷人世界。從最初的定理開始，結果的優雅和廣泛性就顯而易見。起點是將最初僅針對實數參數給定的函數擴展到在複數參數下定義的簡單想法。從那裡，我們進一步探討全純函數的主要性質，其證明通常簡短而具有啟發性：柯西定理、殘差、解析延拓和論證原理。

在這個背景下，讀者將準備好學習大量將這一主題與數學的其他領域相關聯的額外材料：用積分路徑來處理傅立葉變換、黎曼點和素數定理，以及橢圓函數的介紹，最終應用於組合學和數論。

《複分析》全面發展了這個具有多方面影響的主題，同時在概念洞察和嚴謹分析的技術基礎之間取得了平衡。這本書將受到數學、物理、工程和其他科學領域的學生的歡迎。

《普林斯頓分析講座》是一個持續努力，旨在介紹數學分析的核心領域，同時展示它們之間的有機統一性。計劃中的四卷書中的許多例子和應用，其中《複分析》是第二卷，突顯了分析中某些思想對其他數學領域和各種科學領域的深遠影響。斯坦和沙卡爾奇從介紹傅立葉級數和積分開始，深入探討了複分析、測度和積分理論、希爾伯特空間，最後是功能分析、分布和概率論的進一步主題。