# Book:J.C. Burkill/A Second Course in Mathematical Analysis

## J.C. Burkill and H. Burkill: A Second Course in Mathematical Analysis

Published $1970$, Cambridge University Press

ISBN 0 521 28061 3.

### Contents

Preface
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. METRIC 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 $O$, $o$, $\sim$. 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 generalisation. Notes.
6. INTEGRATION
6.1. The Riemann-Stieltjes integral. 6.2. Further properties of the Riemann-Stieltjes integral. 6.3. Improper Riemann-Stieltjes integrals. 6.4. Functions of bounded variation. 6.5. Integrities of bounded variation. 6.6. The Riesz representation theorem. 6.7. The Riemann integral. 6.8. 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 and 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. 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. Cesàro 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 $z = \infty$. 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