Complex variables

CHAPTER ONE Complex numbers 
1.1. Real numbers 
1.2. Complex numbers 
1.3. Complex numbers as marks in a plane 
1.4. Complex numbers as vectors in a plane 
1.5. Addition and subtraction 
1.6. Multiplication and division 
1.7. Summary and notation 
1.8. Conjugate numbers 
1.9. Vectorial operations 
1.10. Limits 
Additional examples and comments on Chapter 1 
CHAPTER TWO Complex functions 
2.1. Extension to the complex domain 
2.2. Exponential function 
2.3. Trigonometric functions 
2.4. Consequences of Euler\'s theorem 
2.5. Further applications of Euler\'s theorem 
2.6. Logarithms 
2.7. Powers 
2.8. Inverse trigonometric functions 
2.9. General remarks 
2.10. Complex function of a real variable: kinematic representation 
2.11. Real functions of a complex variable: graphical representation 
2.12. Complex functions of a complex variable: graphical representation on two planes 
2.13. Complex functions of a complex variable: physical representation in one plane 
Additional examples and comments on Chapter 2 
CHAPTER THREE Differentiation: analytic functions 
3.1. Derivatives 
3.2. Rules for differentiation 
3.3. Analytic condition for differentiability: the Cauchy- Riemann equations 
3.4. Graphical interpretation of differentiability: conformal mapping 
3.5. Physical interpretation of differentiability: sourceless and irrotational vector-fields 
3.6. Divergence and curl 
3.7. Laplace\'s equation 
3.8. Analytic functions 
3.9. Summary and outlook 
Additional examples and comments on Chapter 3 
CHAPTER FOUR Conformal mapping by given functions 
4.1. The stereographic or Ptolemy projection 
4.2. Properties of the stereo graphic projection 
4.3. The bilinear transformation 
4.4. Properties of the bilinear transformation 
4.5. The transformation w = z^2 
4.6. The transformation w = e^z 
4.7. The Mercator map 
Additional examples and comments on Chapter 4 
CHAPTER FIVE Integration: Cauchy\'s theorem 
5.1. Work and flux 
5.2. The main theorem 
5.3. Complex line integrals 
5.4. Rules for integration 
5.5. The divergence theorem 
5.6. A more formal proof of Cauchy\'s theorem 
5.7. Other forms of Cauchy\'s theorem 
5.8. The indefinite integral in the complex domain 
5.9. Geometric language 
Additional examples and comments on Chapter 5 
CHAPTER SIX Cauchy\'s integral formula and applications 
6.1. Cauchy\'s integral formula 
6.2. A first application to the evaluation of definite integrals 
6.3. Some consequences of the Cauchy formula: higher derivatives 
6.4. More consequences of the Cauchy formula: the principle of maximum modulus 
6.5. Taylor\'s theorem, MacLaurin\'s theorem 
6.6. Laurent\'s theorem 
6.7. Singularities of analytic functions 
6.8. The residue theorem 
6.9. Computation of residues 
6.10. Evaluation of definite integrals 
Additional examples and comments on Chapter 6 
CHAPTER SEVEN Conformal mapping and analytic continuation 
7.1. Analytic continuation 
7.2. The gamma function 
7.3. Schwarz\' reflection principle 
7.4. The general mapping problem: Riemann\'s mapping theorem 
7.5. The Schwarz-Christoffei mapping 
7.6. A discussion of the Schwarz-Christoffel formula 
7.7. Degenerate polygons 
Additional examples and comments on Chapter 7 
CHAPTER EIGHT Hydrodynamics 
8.1. The equations of hydrodynamics 
8.2. The complex potential 
8.3. Flow in channels: sources, sinks, and dipoles 
8.4. Flow in channels: conformal mapping 
8.5. Flows past fixed bodies 
8.6. Flows with free boundaries 
CHAPTER NINE Asymptotic expansions 
9.1. Asymptotic series 
9.2. Notation and definitions 
9.3. Manipulating asymptotic series 
9.4. Laplace\'s asymptotic formula 
9.5. Perron\'s extension of Laplace\'s formula 
9.6. The saddle-point method 
Additional examples and comments on Chapter 9 
Index