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174 pages

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en

The theory of complex variables is significant in pure mathematics, and the basis for important applications in applied mathematics.

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Content

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. Applications to applied mathematics are omitted, although Fourier transforms are mentioned. The first part is based on an introductory lecture course, and the second expands on the methods used for the evaluation of real integrals. Numerous worked examples are provided throughout.

- Part I: An introduction to complex variables
- Preface
- Introduction
- Complex Numbers
- Elementary properties
- Inequalities
- Roots
- Exercises 1

- Functions
- Elementary functions
- Exercises 2

- Differentiability
- Definition
- The derivative in detail
- Analyticity
- Harmonic functions
- Exercises 3

- Integration in the complex plane
- The line integral
- The fundamental theorem of calculus
- Closed contours
- Exercises 4

- The Integral Theorems
- Cauchy’s Integral Theorem (1825)
- Cauchy’s Integral Formula (1831)
- An integral inequality
- An application to the evaluation of real integrals
- Exercises 5

- Power Series
- The Laurent expansion (1843)
- Exercises 6

- The Residue Theorem
- The (Cauchy) Residue Theorem (1846)
- Application to real integrals
- Using a different contour
- Exercises 7

- The Fourier Transform
- FTs of derivatives
- Exercises 8
- Answers

- Part II: The integral theorems of complex analysis with applications to the evaluation of real integrals
- List of Integrals
- Preface
- Introduction
- Complex integration
- Exercises 1

- The integral theorems
- Green’s theorem
- Cauchy’s integral theorem
- Cauchy’s integral formula
- The (Cauchy) residue theorem
- Exercises 2

- Evaluation of simple, improper real integrals
- Estimating integrals on semi-circular arcs
- Real integrals of type 1
- Real integrals of type 2
- Exercises 3

- Indented contours, contours with branch cuts and other special contours
- Cauchy principal value
- The indented contour
- Contours with branch cuts
- Special contours
- Exercises 4

- Integration of rational functions of trigonometric functions
- Exercises 5
- Answers

- Biographical Notes
- Index