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.
Author(s): J. C. Burkill
Edition: 1
Publisher: Cambridge University Press
Year: 1970
Language: English
Pages: 534
1. SETS AND FUNCTIONS
1.1 Sets and numbers
1.2. Ordered pairs and Cartesian products
1.3. Functions
1.4. Similarity of sets
Notes
2. METRTC SPACES
2.1. Metrics
2.2. Norms
2.3. Open and closed sets
Notes
3. CONTINUOUS FUNCTIONS ON METRIC SPACES
3.1. Limits
3.2. Continuous functions
3.3. Connected metric spaces
3.4. Complete metric spaces
3.5. Completion of metric spaces
3.6. Compact metric spaces
3.7. The Heine-Borel theorem
Notes
4. LIMITS IN THE SPACES R^1 AND Z
4.1. The symbols 0, o, ~
4.2. Upper and lower limits
4.3. Series of complex terms
4.4. Series of positive terms
4.5. Conditionally convergent real series
4.6. Power series
4.7. Double and repeated limits
Notes
5. UNIFORM CONVERGENCE
5.1. Pointwise and uniform convergence
5.2. Properties assured by uniform convergence
5.3. Criteria for uniform convergence
5.4. Further properties of power series
5.5. Two constructions of continuous functions
5.6. Weierstrass's approximation theorem and its generalization
Notes
6. INTEGRATION
6.1. The ruemann-Stieltjes integral
6.2. Further properties of the Riemann-Stieltjes integral
6.3. Improper Riemann-Stieltjcs integrals
6.4. Functions of bounded variation
6.5. Integrators of bounded variation
6.6. The Riesz representation theorem
6.7. The Riemann integral
6.S. Content
6.9. Some manipulative theorems
Notes
7. FUNCTIONS FROM R^m TO R^n
7.1. Differentiation
7.2. Operations on differentiable functions
7.3. Some properties of differentiable functions
7.4. The implicit function theorem
7.5. The inverse function theorem
7.6. Functional dependence
7.7. Maxima and minima
7.8. Second nnd higher derivatives
Notes
8. INTEGRALS IN R^n
8.1. Curves
8.2. Line integrals
8.3. Integration over intervals in R^n
8.4. Integration over arbitrary bounded sets in R^n
8.5. Differentiation and integration
8.6. Transformation of integrals
8.7. Functions defined by integrals
Notes
9. FOURIER SERIES
9.1. Trigonometric series, p
9.2. Some special series
9.3. Theorems of Riemann. Dirichlet's integral
9.4. Convergence of Fourier series
9.5. Divergence of Fourier series
9.6. Cesaro and Abel summability of series
9.7. Summability of Fourier series
9.8. Mean square approximation. Parseval's theorem
9.9. Fourier integrals
Notes
10. COMPLEX FUNCTION THEORY
10.1. Complex numbers and functions
10.2. Regular functions
10.3. Conformal mapping
10.4. The bilinear mapping. The extended plane
10.5. Properties of bilinear mappings
10.6. Exponential and logarithm
Notes
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
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
11.8. Cauchy's integral formula
11.9. Successive derivatives of a regular function
Notes
12. EXPANSIONS. SINGULARITIES. RESIDUES
12.1. Taylor's series. Uniqueness of regular functions
12.2. Inequalities for coefficients. Liouville's theorem
12.3. Laurent's series
12.4. Singularities
12.5. Residues
12.6. Counting zeros and poles
12.7. The value z=∞
12.8. Behaviour near singularities
Notes
13. GENERAL THEOREMS. ANALYTIC FUNCTIONS
13.1. Regular functions represented by series or integrals
13.2. Local mappings
13.3. The Weierstrass approach. Analytic continuation
13.4. Analytic functions
Notes
14. APPLICATIONS TO SPECIAL FUNCTIONS
14.1. Evaluation of real integrals by residues
14.2. summation of series by residues
14.3. Partial fractions of cot z
14.4. Infinite products
14.5. The factor theorem of Weierstrass. The sine product
14.6. The gamma function
14.7. Integrals expressed in gamma functions
14.8. Asymptotic formulae
Notes
Solutions of Exercises
References
Index
