A Second Course in Analysis

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