An introduction to the theory of complex variables
Publisher: Bookboon.com
Published: 2013
ISBN-13: 9788740301625
Table of contents: title: Content; title: 1. Part I: An introduction to complex variables; title: 2. Preface; title: 3. Introduction; title: 4. Complex Numbers; title: 4.1. Elementary properties; title: 4.2. Inequalities; title: 4.3. Roots; title: 4.4. Exercises 1; title: 5. Functions; title: 5.1. Elementary functions; title: 5.2. Exercises 2; title: 6. Differentiability; title: 6.1. Definition; title: 6.2. The derivative in detail; title: 6.3. Analyticity; title: 6.4. Harmonic functions; title: 6.5. Exercises 3; title: 7. Integration in the complex plane; title: 7.1. The line integral; title: 7.2. The fundamental theorem of calculus; title: 7.3. Closed contours; title: 7.4. Exercises 4; title: 8. The Integral Theorems; title: 8.1. Cauchy’s Integral Theorem (1825); title: 8.2. Cauchy’s Integral Formula (1831); title: 8.3. An integral inequality; title: 8.4. An application to the evaluation of real integrals; title: 8.5. Exercises 5; title: 9. Power Series; title: 9.1. The Laurent expansion (1843); title: 9.2. Exercises 6; title: 10. The Residue Theorem; title: 10.1. The (Cauchy) Residue Theorem (1846); title: 10.2. Application to real integrals; title: 10.3. Using a different contour; title: 10.4. Exercises 7; title: 11. The Fourier Transform; title: 11.1. FTs of derivatives; title: 11.2. Exercises 8; title: 11.3. Answers; title: 12. Part II: The integral theorems of complex analysis with applications to the evaluation of real integrals; title: 13. List of Integrals; title: 14. Preface; title: 15. Introduction; title: 16. Complex integration; title: 16.1. Exercises 1; title: 17. The integral theorems; title: 17.1. Green’s theorem; title: 17.2. Cauchy’s integral theorem; title: 17.3. Cauchy’s integral formula; title: 17.4. The (Cauchy) residue theorem; title: 17.5. Exercises 2; title: 18. Evaluation of simple, improper real integrals; title: 18.1. Estimating integrals on semi-circular arcs; title: 18.2. Real integrals of type 1; title: 18.3. Real integrals of type 2; title: 18.4. Exercises 3; title: 19. Indented contours, contours with branch cuts and other special contours; title: 19.1. Cauchy principal value; title: 19.2. The indented contour; title: 19.3. Contours with branch cuts; title: 19.4. Special contours; title: 19.5. Exercises 4; title: 20. Integration of rational functions of trigonometric functions; title: 20.1. Exercises 5; title: 20.2. Answers; title: 21. Biographical Notes; title: 22. Index