Limits: A New Approach to Real Analysis

ISBN-13: 9781461268727 / Angielski / Miękka / 2012 / 190 str.

Alan F. Beardon; Alan F. Beardon

Opis

Broadly speaking, analysis is the study of limiting processes such as sum- ming infinite series and differentiating and integrating functions, and in any of these processes there are two issues to consider; first, there is the question of whether or not the limit exists, and second, assuming that it does, there is the problem of finding its numerical value. By convention, analysis is the study oflimiting processes in which the issue of existence is raised and tackled in a forthright manner. In fact, the problem of exis- tence overshadows that of finding the value; for example, while it might be important to know that every polynomial of odd degree has a zero (this is a statement of existence), it is not always necessary to know what this zero is (indeed, if it is irrational, we may never know what its true value is). Despite the fact that this book has much in common with other texts on analysis, its approach to the subject differs widely from any other text known to the author. In other texts, each limiting process is discussed, in detail and at length before the next process. There are several disadvan- tages in this approach. First, there is the need for a different definition for each concept, even though the student will ultimately realise that these different definitions have much in common.

Spis treści

I Foundations.- 1 Sets and Functions.- 1.1 Sets.- 1.2 Ordered pairs.- 1.3 Functions.- 2 Real and Complex Numbers.- 2.1 Algebraic properties of real numbers.- 2.2 Order.- 2.3 Upper and lower bounds.- 2.4 Complex numbers.- 2.5 Notation.- II Limits.- 3 Limits.- 3.1 Introduction.- 3.2 Directed sets.- 3.3 The definition of a limit.- 3.4 Examples of limits.- 3.5 Sums, products, and quotients of limits.- 3.6 Limits and inequalities.- 3.7 Functions tending to infinity.- 4 Bisection Arguments.- 4.1 Nested intervals.- 4.2 The Intermediate Value Therem.- 4.3 The Mean Value Inequality.- 4.4 The Cauchy Criterion.- 5 Infinite Series.- 5.1 Infinite series.- 5.2 Unordered sums.- 5.3 Absolute convergence and rearrangements.- 5.4 The Cauchy Product.- 5.5 Iterated sums.- 6 Periodic Functions.- 6.1 The exponential function.- 6.2 The trigonometric functions.- 6.3 Periodicity and ?.- 6.4 The argument of a complex number.- 6.5 The logarithm.- III Analysis.- 7 Sequences.- 7.1 Convergent sequences.- 7.2 Some important examples.- 7.3 Bounded sequences.- 7.4 The Fundamental Theorem of Algebra.- 7.5 Unbounded sequences.- 7.6 Upper and lower limits.- 8 Continuous Functions.- 8.1 Continuous functions.- 8.2 Functions continuous on an interval.- 8.3 Monotonic functions.- 8.4 Uniform continuity.- 8.5 Uniform convergence.- 9 Derivatives.- 9.1 The derivative.- 9.2 The Chain Rule.- 9.3 The Mean Value Theorem.- 9.4 Inverse functions.- 9.5 Power series.- 9.6 Taylor series.- 10 Integration.- 10.1 The integral.- 10.2 Upper and lower integrals.- 10.3 Integrable functions.- 10.4 Integration and differentiation.- 10.5 Improper integrals.- 10.6 Integration and differentiation of series.- 11 ?, ?, e, and n!.- 11.1 The number e.- 11.2 The number ?.- 11.3 Euler’s constant ?.- 11.4 Stirling’s formula for n!.- 11.5 A series and an integral for ?.- Appendix: Mathematical Induction.- References.