A Second Course in Analysis

✍ Scribed by J. C. Burkill

• Publisher: Cambridge University Press
• Year: 1970
• Tongue: English
• Leaves: 532
• Edition: 1
• 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

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