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Examples and Theorems in Analysis / by Peter Walker

Edition 1st ed. 2004.
Publisher (London : Springer London : Imprint: Springer)
Year 2004
Size X, 287 p : online resource
Authors *Walker, Peter author
SpringerLink (Online service)
Subjects LCSH:Mathematical analysis
LCSH:Fourier analysis
LCSH:Functions of real variables
FREE:Analysis
FREE:Fourier Analysis
FREE:Real Functions
Notes 1. Sequences -- 1.1 Examples, Formulae and Recursion -- 1.2 Monotone and Bounded Sequences -- 1.3 Convergence -- 1.4 Subsequences -- 1.5 Cauchy Sequences -- Exercises -- 2. Functions and Continuity -- 2.1 Examples -- 2.2 Monotone and Bounded Functions -- 2.3 Limits and Continuity -- 2.4 Bounds and Intermediate Values -- 2.5 Inverse Functions -- 2.6 Recursive Limits and Iteration -- 2.7 One-Sided and Infinite Limits. Regulated Functions -- 2.8 Countability -- Exercises -- 3. Differentiation -- 3.1 Differentiable Functions -- 3.2 The Significance of the Derivative -- 3.3 Rules for Differentiation -- 3.4 Mean Value Theorems and Estimation -- 3.5 More on Iteration -- 3.6 Optimisation -- Exercises -- 4. Constructive Integration -- 4.1 Step Functions -- 4.2 The Integral of a Regulated function -- 4.3 Integration and Differentiation -- 4.4 Applications -- 4.5 Further Mean Value Theorems -- Exercises -- 5. Improper Integrals -- 5.1 Improper Integrals on an Interval -- 5.2 Improper Integrals at Infinity -- 5.3 The Gamma function -- Exercises -- 6. Series -- 6.1 Convergence -- 6.2 Series with Positive Terms -- 6.3 Series with Arbitrary Terms -- 6.4 Power Series -- 6.5 Exponential and Trigonometric Functions -- 6.6 Sequences and Series of Functions -- 6.7 Infinite Products -- Exercises -- 7. Applications -- 7.1 Fourier Series -- 7.2 Fourier Integrals -- 7.3 Distributions -- 7.4 Asymptotics -- 7.5 Exercises -- A. Fubini’s Theorem -- B. Hints and Solutions for Exercises
Examples and Theorems in Analysis takes a unique and very practical approach to mathematical analysis. It makes the subject more accessible by giving the examples equal status with the theorems. The results are introduced and motivated by reference to examples which illustrate their use, and further examples then show how far the assumptions may be relaxed before the result fails. A number of applications show what the subject is about and what can be done with it; the applications in Fourier theory, distributions and asymptotics show how the results may be put to use. Exercises at the end of each chapter, of varying levels of difficulty, develop new ideas and present open problems. Written primarily for first- and second-year undergraduates in mathematics, this book features a host of diverse and interesting examples, making it an entertaining and stimulating companion that will also be accessible to students of statistics, computer science and engineering, as well as to professionals in these fields
HTTP:URL=https://doi.org/10.1007/978-0-85729-380-0
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Springer eBooks 9780857293800
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EB00203097

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Material Type E-Book
Classification LCC:QA299.6-433
DC23:515
ID 4000104694
ISBN 9780857293800

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