Numerical+Methods.pdf
Numerical Analysis for almost four decades at the Indian Institute of Technology New Delhi. and co-author of their earlier books on Numerical Analysis.
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An Introduction to Numerical Methods and Analysis
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PRINCIPLES OF MATHEMATICAL ANALYSIS
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Numerical analysis is a branch of Mathematics that deals with devising e?cient methods for obtaining numerical solutions to di?cult Mathematical problems Most of the Mathematical problems that arise in science and engineering are very hard and sometime impossible to solve exactly
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INTERNATIONAL SERIES IN PURE AND APPLIED MATHEMATICS
Principles of mathematical analysis (International series in pure and applied mathematics) Bibliography: p Includes index 1 Mathematical analysis I Title QA300 R8 1976 515 75-17903 ISBN 0-07-054235-X PRINCIPLES OF MATHEMATICAL ANALYSIS • Copyright © 1964 1976 by McGraw-Hill Inc Al] rights reserved Copyright 1953 by McGraw-Hill Inc
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Numerical Analysis First Edition Rostam K Saeed Department of Mathematics College of Science Salahaddin University/Erbil Kurdistan Region Hawler Iraq Karwan H F Jwamer Department of
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What is numerical analysis?
- Introduction Numerical analysis is a branch of Mathematics that deals with devising e?cient methods for obtaining numerical solutions to di?cult Mathematical problems. Most of the Mathematical problems that arise in science and engineering are very hard and sometime impossible to solve exactly.
What are the three parts of numerical analysis?
- Numerical analysis include three parts. The ?rst part of the subject is about the development of a method to a problem. The second part deals with the analysis of the method, which includes the error analysis and the e?ciency analysis.
Who invented the equivalence class of rational numbers?
- in which each real number is defined to be an equivalence class of Cauchy sequences of rational numbers (see Chap. 3), is carried out in Sec. 5 of the book by Hewitt and Stromberg. The cuts in Q which we used here were invented by Dedekind. The construction of R from Q by means of Cauchy sequences is due to Cantor.
PRINCIPLES OF
MATHEMATICAL ANALYSIS
INTERNATIONAL SERIES IN PURE
AND APPLIED MATHEMATICS
William Ted Martin, E. H. Spanier, G. Springer andP. J. Davis. Consulting Editors
AHLFORS: Complex Analysis
BucK: Advanced Calculus
BUSACKER AND SAATY: Finite Graphs and Networks
CHENEY: Introduction to Approximation Theory
CHESTER: Techniques in Partial Differential Equations CODDINGTON AND LEVINSON: Theory of Ordinary Differential Equations CONTE AND DE BooR: Elementary Numerical Analysis: An Algorithmic Approach DENNEMEYER: Introduction to Partial Differential Equations and Boundary ValueProblems
DETTMAN: Mathematical Methods in Physics and Engineering GOLOMB AND SHANKS: Elements of Ordinary Differential Equations GREENSPAN: Introduction to Partial Differential Equations HAMMING: Numerical Methods for Scientists and EngineersHILDEBRAND: Introduction to Numerical Analysis
HousEHOLDER: The Numerical Treatment of a Single Nonlinear Equation KALMAN, FALB, AND ARBIB: Topics in Mathematical Systems TheoryLASS: Vector and Tensor Analysis
McCARTY: Topology: An Introduction with Applications to Topological GroupsMONK: Introduction to Set Theory
MOORE: Elements of Linear Algebra and Matrix TheoryMosTOW AND SAMPSON: Linear Algebra
MouRSUND AND DURIS: Elementary Theory and Application of Numerical AnalysisPEARL: Matrix Theory and Finite Mathematics
PIPES AND HARVILL: Applied Mathematics for Engineers and PhysicistsRALSTON: A First Course in Numerical Analysis
RITGER AND RosE: Differential Equations with ApplicationsRITT: Fourier Series
RUDIN: Principles of Mathematical Analysis
SHAPIRO: Introduction to Abstract Algebra
SIMMONS: Differential Equations with Applications and Historical Notes SIMMONS: Introduction to Topology and Modern Analysis SNEDDON: Elements of Partial Differential EquationsSTRUBLE: Nonlinear Differential Equations
McGraw-Hill, Inc.
New York St. Louis San Francisco Auckland Bogota
Caracas Lisbon London Madrid Mexico City Milan
Montreal New Delhi San Juan Singapore
Sydney Tokyo Toronto
WALTER RUDIN
Professor of Mathematics
University
of Wisconsin,-MadisonTHIRD EDITION
This book was set in Times New Roman.
The editors were A. Anthony Arthur and Shelly Levine Langman; the production supervisor was Leroy A. Young. R. R. Donnelley & Sons Company was printer and binder.This book is printed on acid-free paper.
Library of Congress Cataloging in Publication DataRudin, Walter, date
Principles of mathematical analysis.
(International series in pure and applied mathematics)Bibliography: p.
Includes index.
1. Mathematical analysis. I. Title.
QA300.R8 1976 515 75-17903
ISBN 0-07-054235-X
PRINCIPLES OF MATHEMATICAL ANALYSIS
Copyright © 1964, 1976 by McGraw-Hill, Inc. Al] rights reserved.Copyright
1953 by McGraw-Hill, Inc. All rights reserved.
Printed
in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.28 29 30 DOC/DOC O 9 8 7 6 5 4 3 2 1 0
Preface
Chapter 1 The Real and Complex Number Systems
Introduction
Ordered Sets
Fields
The Real Field
The Extended Real Number System
The Complex Field
Euclidean Spaces
Appendix
Exercises
Chapter 2 Basic Topology
Finite, Countable, and Uncountable Sets
Metric Spaces
Compact Sets
Perfect Sets
CONTENTS
lX 1 1 3 5 8 11 12 16 17 2124
24
30
36
41
Vi CONTENTS
Connected Sets
Exercises
Chapter 3 Numerical Sequences and Series
Convergent Sequences
Subsequences
Cauchy Sequences
Upper and Lower Limits
Some Special Sequences
Series
Series
of Nonnegative TermsThe Number
eThe Root and Ratio Tests
Power Series
Summation
by PartsAbsolute Convergence
Addition and Multiplication of Series
Rearrangements
Exercises
Chapter 4 Continuity
Limits of Functions
Continuous Fur1ctions
Continuity and Compactness
Continuity and Connectedness
Discontinuities
Monotonic Functions
Infinite Limits and Limits at Infinity
Exercises
Chapter 5 Differentiation
The Derivative of a Real Function
Mean Value Theorems
The Continuity of Derivatives
L'Hospital's Rule
Derivatives
of Higher OrderTaylor's Theorem
Differentiation of Vector-valued Functions
Exercises
4243
47
47
51
52
55
57
58
61
63
65
69
70
71
72
75
78
83
83
85
89
93
94
95
97
98
103
103
107
108
109
110
110
111
114
Chapter 6 The Riemann-Stieltjes Integral
Definition
and Existence of the IntegralProperties
of the IntegralIntegration
and DifferentiationIntegration
of Vector-valued FunctionsRectifiable Curves
Exercises
Chapter 7 Sequences
and Series of Functions.Discussion
of Main ProblemUniform Convergence
Uniform Convergence and Continuity
Uniform Convergence and Integration
Uniform Convergence and Differentiation
Equicontinuous Families
of FunctionsThe Stone-Weierstrass Theorem
Exercises
Chapter 8 Some Special Functions
Power Series
The Exponential
and Logarithmic FunctionsThe Trigonometric Functions
The Algebraic Completeness of the Complex Field
Fourier Series
TheGamma Function
Exercises
Chapter 9 Functions
of Several VariablesLinear
Transformations
Differentiation
The Contraction Principle
The Inverse Function Theorem
The Implicit Function Theorem
TheRank Theorem
Determinants
Derivatives of Higher Order
Differentiation of Integrals
Exercises
Chapter
10 Integration of Differential Forms
Integration
CONTENTS vii
120120
128
133
135
136
138
143
143
147
149
151
152
154
159
165
172
172
178
182
184
185
192
196
204
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