A Second Course in Mathematical Analysis

✍ Scribed by J. C. Burkill, H. Burkill

• Publisher: Cambridge University Press
• Year: 2002
• Tongue: English
• Leaves: 538
• Series: Cambridge Mathematical Library
• Edition: New
• Category: Library

✦ Synopsis

The classic analysis textbook from Burkill and Burkill is now available in the Cambridge Mathematical Library. This straightforward course, based on the idea of a limit, is for students of mathematics and physics who have acquired a working knowledge of calculus and are ready for a more systematic approach. The treatment given here also brings in other limiting processes, such as the summation of infinite series and the expansion of trigonometric functions as power series. Particular attention is given to clarity of exposition and the logical development of the subject matter.

✦ Table of Contents

CONTENTS
PREFACE
1 SETS AND FUNCTIONS
1.1. Sets and numbers
Exercises 1(a)
Solutions 1(a)
1.2. Ordered pairs and Cartesian products
1.3. Functions
Exercises 1(b)
Solution 1(b)
1.4. Similarity of sets
Exercises 1(c)
Solution 1(c)
NOTES ON CHAPTER 1
2 METRIC SPACES
2.1. Metrics
Exercises 2(a)
Solutions 2(a)
2.2. Norms
Exercises 2(b)
Solutions 2(b)
2.3. Open and closed sets
Exercises 2(c)
Solutions 2(c)
NOTES ON CHAPTER 2
3 CONTINUOUS FUNCTIONS ON METRIC SPACES
3.1. Limits
Exercises 3(a)
Solutions 3(a)
3.2. Continuous functions
Exercises 3(b)
Solutions 3(b)
3.3. Connected metric spaces
Exercises 3(c)
Solutions 3(c)
3.4. Complete metric spaces
Exercises 3(d)
Solutions 3(d)
3.5. Completion of metric spaces
Exercises 3(e)
Solutions 3(e)
3.6. Compact metric spaces
Exercises 3(f)
Solutions 3(f)
3.7. The Heine-Borel theorem
Exercises 3(g)
Solutions 3(g)
NOTES ON CHAPTER 3
4 LIMITS IN THE SPACES R^1 AND Z
4.1. The symbols O, o, ~
Exercises 4(a)
Solutions 4(a)
4.2. Upper and lower limits
Exercises 4(b)
Solutions 4(b)
4.3. Series of complex terms
Exercises 4(c)
Solutions 4(c)
4.4. Series of positive terms
Exercises 4(d)
Solutions 4(d)
4.5. Conditionally convergent real series
Exercises 4(e)
Solutions 4(e)
4.6. Power series
Exercises 4(f)
Solutions 4(f)
4.7. Double and repeated limits
Exercises 4(g)
Solutions 4(g)
NOTES ON CHAPTER 4
5 UNIFORM CONVERGENCE
5.1. Pointwise and uniform convergence
Exercises 5(a)
Solutions 5(a)
5.2. Properties assured by uniform convergence
Exercises 5(b)
Solutions 5(b)
5.3. Criteria for uniform convergence
Exercises 5(c)
Solutions 5(c)
5.4. Further properties of power series
Exercises 5(d)
Solutions 5(d)
5.5. Two constructions of continuous functions
Exercises 5(e)
Solutions 5(e)
5.6. Weierstrass’s approximation theorem and its generalization
Exercises 5(f)
Solutions 5(f)
NOTES ON CHAPTER 5
6 INTEGRATION
6.1. The Riemann-Stieltjes integral
Exercises 6(a)
Solutions 6(a)
6.2. Further properties of the Riemann-Stieltjes integral
Exercises 6(b)
Solutions 6(b)
6.3. Improper Riemann-Stieltjes integrals
Exercises 6(c)
Solutions 6(c)
6.4. Functions of bounded variation
Exercises 6(d)
Solutions 6(d)
6.5. Integrators of bounded variation
Exercises 6(e)
Solutions 6(e)
6.6. The Riesz representation theorem
Exercises 6(f)
Solutions 6(f)
6.7. The Riemann integral
Exercises 6(g)
Solutions 6(g)
6.8. Content
Exercises 6(h)
Solutions 6(h)
6.9. Some manipulative theorems
Exercises 6(i)
Solutions 6(i)
NOTES ON CHAPTER 6
7 FUNCTIONS FROM R^m TO R^n
7.1. Differentiation
Exercises 7(a)
Solutions 7(a)
7.2. Operations on differentiable functions
Exercises 7(b)
Solutions 7(b)
7.3. Some properties of differentiable functions
Exercises 7(c)
Solutions 7(c)
7.4. The implicit function theorem
Exercises 7(d)
Solutions 7(d)
7.5. The inverse function theorem
Exercises 7(e)
Solutions 7(e)
7.6. Functional dependence
Exercises 7(f)
Solutions 7(f)
7.7. Maxima and minima
Exercises 7(g)
Solutions 7(g)
7.8. Second and higher derivatives
Exercises 7(h)
Solutions 7(h)
NOTES ON CHAPTER 7
8 INTEGRALS IN R^n
8.1. Curves
Exercises 8(a)
Solutions 8(a)
8.2. Line integrals
Exercises 8(b)
Solutions 8(b)
8.3. Integration over intervals in R^n
Exercises 8(c)
Solutions 8(c)
8.4. Integration over arbitrary bounded sets in R^n
Exercises 8(d)
Solutions 8(d)
8.5. Differentiation and integration
Exercises 8(e)
Solutions 8(e)
8.6. Transformation of integrals
Exercises 8(f)
Solutions 8(f)
8.7. Functions defined by integrals
Exercises 8(g)
Solutions 8(g)
NOTES ON CHAPTER 8
9 FOURIER SERIES
9.1. Trigonometric series
9.2. Some special series
Exercises 9(a)
Solutions 9(a)
9.3. Theorems of Riemann. Dirichlet’s integral
9.4. Convergence of Fourier series
Exercises 9(b)
Solutions 9(b)
9.5. Divergence of Fourier series
Exercises 9(c)
Solutions 9(c)
9.6. Cesaro and Abel summability of series
9.7. Summability of Fourier series
Exercises 9(d)
Solutions 9(d)
9.8. Mean square approximation. Parseval’s theorem
Exercises 9(e)
Solutions 9(e)
9.9. Fourier integrals
Exercises 9(f)
Solutions 9(f)
NOTES ON CHAPTER 9
10 COMPLEX FUNCTION THEORY
10.1. Complex numbers and functions
10.2. Regular functions
Exercises 10(a)
Solutions 10(a)
10.3. Conformal mapping
10.4. The bilinear mapping. The extended plane
10.5. Properties of bilinear mappings
Exercises 10(b)
Solutions 10(b)
10.6. Exponential and logarithm
Exercises 10(c)
Solutions 10(c)
NOTES ON CHAPTER 10
11 COMPLEX INTEGRALS. CAUCHY’S THEOREM
11.1. Complex integrals
11.2. Dependence of the integral on the path
11.3. Primitives and local primitives
Exercises 11(a)
Solutions 11(a)
11.4, Cauchy’s theorem for a rectangle
11.5. Cauchy’s theorem for circuits in a disc
11.6. Homotopy. The general Cauchy theorem
11.7. The index of a circuit for a point
Exercises 11(b)
Solutions 11(b)
11.8. Cauchy’s integral formula
11.9. Successive derivatives of a regular function
Exercises 11(c)
Solutions 11(c)
NOTES ON CHAPTER 11
12 EXPANSIONS. SINGULARITIES. RESIDUES
12.1. Taylor’s series. Uniqueness of regular functions
12.2. Inequalities for coefficients. Liouville’s theorem
Exercises 12(a)
Solutions 12(a)
12.3 Laurent’s series
Exercises 12(b)
Solutions 12(b)
12.4. Singularities
12.5. Residues
12.6. Counting zeros and poles
Exercises 12(c)
Solutions 12(c)
12.7. The value z = ∞
12.8. Behaviour near singularities
Exercises 12(d)
Solutions 12(d)
NOTES ON CHAPTER 12
13 GENERAL THEOREMS. ANALYTIC FUNCTIONS
13.1. Regular functions represented by series or integrals
Exercises 13(a)
Solutions 13(a)
13.2. Local mappings
Exercises 13(b)
Solutions 13(b)
13.3. The Weierstrass approach. Analytic continuation
13.4. Analytic functions
Exercises 13(c)
Solutions 13(c)
NOTES ON CHAPTER 13
14 APPLICATIONS TO SPECIAL FUNCTIONS
14.1. Evaluation of real integrals by residues
Exercises 14(a)
Solutions 14(a)
14.2. Summation of series by residues
14.3. Partial fractions of cot z
Exercises 14(b)
Solutions 14(b)
14.4 Infinite products
14.5. The factor theorem of Weierstrass. The sine product
Exercises 14(c)
Solutions 14(c)
14.6. The gamma function
Exercises 14(d)
Solutions 14(d)
14.7. Integrals expressed in gamma functions
Exercises 14(e)
Solutions 14(e)
14.8. Asymptotic formulae
NOTES ON CHAPTER 14
SOLUTIONS OF EXERCISES
REFERENCES
INDEX
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NOPQRS
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