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1 online resource (ix, 269 pages) : illustrations
Complex Variables covers topics ranging from complex numbers to point sets in the complex plane, elementary functions, straight lines and circles, simple and conformal transformations, and zeros and singularities. Cauchy's theorem, Taylor's theorem, Laurent's theorem, contour integration, and miscellaneous theorems are also discussed. This volume consists of 14 chapters, the first of which introduces the theory of complex numbers and their development either from an algebraic or from a geometrical viewpoint. Emphasis is on the complex plane, modulus, amplitude, number pairs, complex conjugates.
Front Cover; Complex Variables; Copyright Page; Table of Contents; INTRODUCTION; CHAPTER 1. COMPLEX NUMBERS; 1.1. The Complex Plane; 1.2. Modulus; 1.3. Amplitude; 1.4. Number Pairs; 1.5. Addition; 1.6. Scalar Multiplication; 1.7. Subtraction; 1.8. Multiplication; 1.9. Division; 1.10. An Alternative Notation; 1.11. An Algebraic Approach; 1.12. Complex Numbers as an Extension of the Real Number Field; 1.13. Complex Conjugates; 1.14. The Triangle Inequality; 1.15. De Moivre's Theorem; Exercises; CHAPTER 2. POINT SETS IN THE COMPLEX PLANE. SEQUENCES. LIMITS.
2.1. Point Sets: Finite, Countable, and Non-countable Sets. Real Intervals2.2. Bounded and Unbounded Sets on the Real Line; 2.3. The Bolzano-Weierstrass Property; 2.4. Bounded and Unbounded Sets in the Complex Plane; 2.5. Neighbourhoods. Open Sets; 2.6. Limit Points; 2.7. Closed Sets; 2.8. Boundary Points; 2.9. Closure; 2.10. Sequences; 2.11. Convergence; 2.12. Divergence; 2.13. Boundedness of Convergent Sequences; 2.14. A Test for Convergence; 2.15. Cauchy Sequences of Real Numbers; 2.16. Cauchy Sequences of Complex Numbers; 2.17. Non-decreasing Real Sequences; Exercises.
CHAPTER 3. INFINITE SERIES. TESTS FOR CONVERGENCE3.1. The Sum of an Infinite Series; 3.2. Summability; 3.3. Testing for Convergence or Divergence; 3.4. The Comparison Test; 3.5. d'Alembert's Ratio Test; 3.6. Upper and Lower Limits; 3.7. Cauchy's Root Test; 3.8. The Integral Test; 3.9. Series with Negative or Complex Terms; 3.10. Absolute Convergence; 3.11. Other Tests; 3.12. Multiplication of Series; Exercises; CHAPTER 4. FUNCTIONS OF A COMPLEX VARIABLE; 4.1. The Definition of a Function; 4.2. Continuity; 4.3. Differentiability; 4.4. The Cauchy-Riemann Equations.
4.5. The Cauchy-Riemann Equations. Sufficiency4.6. Analytic Functions; 4.7. Laplace's Equation; 4.8. Orthogonal Families of Curves; Exercises; CHAPTER 5. ELEMENTARY FUNCTIONS; 5.1. Polynomials; 5.2. Rational Functions; 5.3. The Exponential Function; 5.4. Sine and Cosine; 5.5. The Link between the Exponential and Trigonometric Functions; 5.6. de Moivre's Theorem; 5.7. Hyperbolic Functions; 5.8. The Logarithmic Function; 5.9. More General Power Functions; 5.10. The Expression of a Regular Function as a Series; 5.11. Differentiability of Power Series.
5.12. Repeated Differentiation of an Infinite Series5.13. Inverse Functions; Exercises; CHAPTER 6. STRAIGHT LINE AND CIRCLE; 6.1. The Standard Equation of a Straight Line; 6.2. Other Forms of the Equation; 6.3. The Circle; Exercises; CHAPTER 7. SIMPLE TRANSFORMATIONS; 7.1. Translation; 7.2. Reflection; 7.3. Rotation; 7.4. Magnification; 7.5. Glide Reflection; 7.6. Shear; 7.7. Inversion; 7.8. The Point at Infinity; Exercises; CHAPTER 8. CONFORMAL TRANSFORMATIONS; 8.1. Regular Transformations; 8.2. The Bilinear Transformation; 8.3. Straight Lines and Circles; 8.4. The Mapping of a Domain.
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Includes bibliographical references (page 255).