An introduction to the theory of complex variables
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ISBN: 9788740301625
1 edition
Pages : 174
 Price: 129.00 kr
 Price: €13.99
 Price: £13.99
 Price: ₹250
 Price: $13.99
 Price: 129.00 kr
 Price: 129.00 kr
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About the book
Description
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.
Content
 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 semicircular 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