Linear and Complex Analysis for Applications

Linear and Complex Analysis for Applications aims to unify various parts of mathematical analysis in an engaging manner and to provide a diverse and unusual collection of applications, both to other fields of mathematics and to physics and engineering. The book evolved from several of the author's teaching experiences, his research in complex analysis in several variables, and many conversations with friends and colleagues. It has three primary goals:

1. to develop enough linear analysis and complex variable theory to prepare students in engineering or applied mathematics for advanced work,
2. to unify many distinct and seemingly isolated topics,
3. to show mathematics as both interesting and useful, especially via the juxtaposition of examples and theorems.

The book realizes these goals by beginning with reviews of Linear Algebra, Complex Numbers, and topics from Calculus III. As the topics are being reviewed, new material is inserted to help the student develop skill in both computation and theory. The material on linear algebra includes infinite-dimensional examples arising from elementary calculus and differential equations. Line and surface integrals are computed both in the language of classical vector analysis and by using differential forms. Connections among the topics and applications appear throughout the book.

The text weaves abstract mathematics, routine computational problems, and applications into a coherent whole, whose unifying theme is linear systems. It includes many unusual examples and contains more than 450 exercises.

《線性和複雜分析應用》旨在以引人入勝的方式統一數學分析的各個部分，並提供多樣且不尋常的應用，包括其他數學領域以及物理學和工程學。該書源於作者多年的教學經驗、在多變數複雜分析領域的研究以及與朋友和同事的許多對話。它有三個主要目標：

1. 發展足夠的線性分析和複變數理論，以便為工程學或應用數學的學生做進階工作的準備。
2. 統一許多不同且看似孤立的主題。
3. 通過例子和定理的並置，展示數學的趣味和實用性。

該書通過回顧線性代數、複數和微積分III的主題來實現這些目標。在回顧主題的同時，插入新的材料以幫助學生在計算和理論方面發展技能。線性代數的材料包括從基礎微積分和微分方程中產生的無窮維例子。線和曲面積分既可以用傳統向量分析的語言計算，也可以使用微分形式計算。書中始終出現主題之間的聯繫和應用。

該書將抽象數學、常規計算問題和應用融入一個有機整體，其統一主題是線性系統。它包含許多不尋常的例子，並包含450多個練習題。

John P. D'Angelo is Professor of Mathematics at the University of Illinois at Urbana-Champaign. He has published more than sixty research papers in complex analysis in several variables and Cauchy-Riemann geometry. He was awarded the Stefan Bergman prize in 1999 for some of this work. He has been recognized for excellence in his teaching and he is a Fellow of the American Mathematical Society.

John P. D'Angelo是伊利諾伊大學香檳分校的數學教授。他在多變數複分析和柯西-黎曼幾何學領域發表了六十多篇研究論文。他因這些工作於1999年獲得了斯特凡·伯格曼獎。他在教學方面獲得了卓越的認可，並且是美國數學學會的會士。