Logo - springer
Slogan - springer

Mathematics - Analysis | Limits - A New Approach to Real Analysis

Limits

A New Approach to Real Analysis

Beardon, Alan F.

1997, IX, 190 p.

Available Formats:
eBook
Information

Springer eBooks may be purchased by end-customers only and are sold without copy protection (DRM free). Instead, all eBooks include personalized watermarks. This means you can read the Springer eBooks across numerous devices such as Laptops, eReaders, and tablets.

You can pay for Springer eBooks with Visa, Mastercard, American Express or Paypal.

After the purchase you can directly download the eBook file or read it online in our Springer eBook Reader. Furthermore your eBook will be stored in your MySpringer account. So you can always re-download your eBooks.

 
$59.95

(net) price for USA

ISBN 978-1-4612-0697-2

digitally watermarked, no DRM

Included Format: PDF

download immediately after purchase


learn more about Springer eBooks

add to marked items

Hardcover
Information

Hardcover version

You can pay for Springer Books with Visa, Mastercard, American Express or Paypal.

Standard shipping is free of charge for individual customers.

 
$79.95

(net) price for USA

ISBN 978-0-387-98274-8

free shipping for individuals worldwide

usually dispatched within 3 to 5 business days


add to marked items

Softcover
Information

Softcover (also known as softback) version.

You can pay for Springer Books with Visa, Mastercard, American Express or Paypal.

Standard shipping is free of charge for individual customers.

 
$79.95

(net) price for USA

ISBN 978-1-4612-6872-7

free shipping for individuals worldwide

usually dispatched within 3 to 5 business days


add to marked items

  • About this textbook

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.

Content Level » Lower undergraduate

Related subjects » Analysis

Table of contents 

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.

Popular Content within this publication 

 

Articles

Read this Book on Springerlink

Services for this book

New Book Alert

Get alerted on new Springer publications in the subject area of Real Functions.

Additional information