An edition of An introduction to the theory of complex variables
(2013)
An introduction to the theory of complex variables
by R. S. Johnson
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Last edited by Tom Morris
February 4, 2018 | History
The theory of complex variables is significant in pure mathematics, and the basis for important applications in applied mathematics (e.g. fluids). This text provides an introduction to the ideas that are met at university: complex functions, differentiability, integration theorems, with applications to real integrals.
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Publish Date
2013
Publisher
Bookboon.com
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Book Details
Table of Contents
Content
1. Part I: An introduction to complex variables
2. Preface
3. Introduction
4. Complex Numbers
4.1. Elementary properties
4.2. Inequalities
4.3. Roots
4.4. Exercises 1
5. Functions
5.1. Elementary functions
5.2. Exercises 2
6. Differentiability
6.1. Definition
6.2. The derivative in detail
6.3. Analyticity
6.4. Harmonic functions
6.5. Exercises 3
7. Integration in the complex plane
7.1. The line integral
7.2. The fundamental theorem of calculus
7.3. Closed contours
7.4. Exercises 4
8. The Integral Theorems
8.1. Cauchy’s Integral Theorem (1825)
8.2. Cauchy’s Integral Formula (1831)
8.3. An integral inequality
8.4. An application to the evaluation of real integrals
8.5. Exercises 5
9. Power Series
9.1. The Laurent expansion (1843)
9.2. Exercises 6
10. The Residue Theorem
10.1. The (Cauchy) Residue Theorem (1846)
10.2. Application to real integrals
10.3. Using a different contour
10.4. Exercises 7
11. The Fourier Transform
11.1. FTs of derivatives
11.2. Exercises 8
11.3. Answers
12. Part II: The integral theorems of complex analysis with applications to the evaluation of real integrals
13. List of Integrals
14. Preface
15. Introduction
16. Complex integration
16.1. Exercises 1
17. The integral theorems
17.1. Green’s theorem
17.2. Cauchy’s integral theorem
17.3. Cauchy’s integral formula
17.4. The (Cauchy) residue theorem
17.5. Exercises 2
18. Evaluation of simple, improper real integrals
18.1. Estimating integrals on semi-circular arcs
18.2. Real integrals of type 1
18.3. Real integrals of type 2
18.4. Exercises 3
19. Indented contours, contours with branch cuts and other special contours
19.1. Cauchy principal value
19.2. The indented contour
19.3. Contours with branch cuts
19.4. Special contours
19.5. Exercises 4
20. Integration of rational functions of trigonometric functions
20.1. Exercises 5
20.2. Answers
21. Biographical Notes
22. Index
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