Fundamental Mathematical Analysis (Paperback)

Magnus, Robert

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商品描述

This textbook offers a comprehensive undergraduate course in real analysis in one variable. Taking the view that analysis can only be properly appreciated as a rigorous theory, the book recognises the difficulties that students experience when encountering this theory for the first time, carefully addressing them throughout.

Historically, it was the precise description of real numbers and the correct definition of limit that placed analysis on a solid foundation. The book therefore begins with these crucial ideas and the fundamental notion of sequence. Infinite series are then introduced, followed by the key concept of continuity. These lay the groundwork for differential and integral calculus, which are carefully covered in the following chapters. Pointers for further study are included throughout the book, and for the more adventurous there is a selection of "nuggets", exciting topics not commonly discussed at this level. Examples of nuggets include Newton's method, the irrationality of π, Bernoulli numbers, and the Gamma function.

Based on decades of teaching experience, this book is written with the undergraduate student in mind. A large number of exercises, many with hints, provide the practice necessary for learning, while the included "nuggets" provide opportunities to deepen understanding and broaden horizons.

商品描述(中文翻譯)

這本教科書提供了一門全面的大學一元實分析課程。書中認為,只有將分析視為一個嚴謹的理論,才能正確地理解它,並且書中在整本書中都細心地解決了學生初次接觸這個理論時所遇到的困難。

從歷史上看,對實數的精確描述和極限的正確定義使得分析學有了堅實的基礎。因此,本書從這些關鍵思想和序列的基本概念開始。接著介紹了無窮級數,然後是連續性的關鍵概念。這些為微分和積分學奠定了基礎,並在接下來的章節中進行了詳細的介紹。書中還包含了進一步學習的指引,對於更有冒險精神的讀者,還提供了一些「寶石」,這些是在這個層次上不常討論的有趣主題。寶石的例子包括牛頓法、π的無理性、伯努利數和伽瑪函數。

基於數十年的教學經驗,本書以本科生為目標讀者。大量的練習題,其中許多有提示,提供了學習所需的練習機會,而包含的「寶石」則提供了深化理解和拓寬視野的機會。

作者簡介

Robert Magnus was born in the UK and studied mathematics at the Universities of Cambridge and Sussex. He has taught nearly all subjects associated with analysis and has published papers in the areas of bifurcation theory, catastrophe theory, analytic operator functions and non-linear partial differential equations. Since 1977 he has lived and worked in Iceland.

作者簡介(中文翻譯)

Robert Magnus出生於英國,在劍橋大學和薩塞克斯大學學習數學。他教授過與分析相關的幾乎所有科目,並在分歧理論、災難理論、解析算子函數和非線性偏微分方程等領域發表了論文。自1977年以來,他一直居住和工作在冰島。