Author | Howie, John M. author |
---|---|
Title | Complex Analysis [electronic resource] / by John M. Howie |
Imprint | London : Springer London : Imprint: Springer, 2003 |
Connect to | http://dx.doi.org/10.1007/978-1-4471-0027-0 |
Descript | XI, 260 p. online resource |
1. What Do I Need to Know? -- 1.1 Set Theory -- 1.2 Numbers -- 1.3 Sequences and Series -- 1.4 Functions and Continuity -- 1.5 Differentiation -- 1.6 Integration -- 1.7 Infinite Integrals -- 1.8 Calculus of Two Variables -- 2. Complex Numbers -- 2.1 Are Complex Numbers Necessary? -- 2.2 Basic Properties of Complex Numbers -- 3. Prelude to Complex Analysis -- 3.1 Why is Complex Analysis Possible? -- 3.2 Some Useful Terminology -- 3.3 Functions and Continuity -- 3.4 The O and o Notations -- 4. Differentiation -- 4.1 Differentiability -- 4.2 Power Series -- 4.3 Logarithms -- 4.4 Cuts and Branch Points -- 4.5 Singularities -- 5. Complex Integration -- 5.1 The Heine-Borel Theorem -- 5.2 Parametric Representation -- 5.3 Integration -- 5.4 Estimation -- 5.5 Uniform Convergence -- 6. Cauchyโs Theorem -- 6.1 Cauchyโs Theorem: A First Approach -- 6.2 Cauchyโs Theorem: A More General Version -- 6.3 Deformation -- 7. Some Consequences of Cauchyโs Theorem -- 7.1 Cauchyโs Integral Formula -- 7.2 The Fundamental Theorem of Algebra -- 7.3 Logarithms -- 7.4 Taylor Series -- 8. Laurent Series and the Residue Theorem -- 8.1 Laurent Series -- 8.2 Classification of Singularities -- 8.3 The Residue Theorem -- 9. Applications of Contour Integration -- 9.1 Real Integrals: Semicircular Contours -- 9.2 Integrals Involving Circular Functions -- 9.3 Real Integrals: Jordanโs Lemma -- 9.4 Real Integrals: Some Special Contours -- 9.5 Infinite Series -- 10. Further Topics -- 10.1 Integration of f?/f; Rouchรฉโs Theorem -- 10.2 The Open Mapping Theorem -- 10.3 Winding Numbers -- 11. Conformai Mappings -- 11.1 Preservation of Angles -- 11.2 Harmonic Functions -- 11.3 Mรถbius Transformations -- 11.4 Other Transformations -- 12. Final Remarks -- 12.1 Riemannโs Zeta function -- 12.2 Complex Iteration -- 13. Solutions to Exercises -- Subject IndexBibliography -- Subject IndexIndex