Complex Analysis

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. Cauchy's Theorem.- 6.1 Cauchy's Theorem: A First Approach.- 6.2 Cauchy's Theorem: A More General Version.- 6.3 Deformation.- 7. Some Consequences of Cauchy's Theorem.- 7.1 Cauchy's 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: Jordan's Lemma.- 9.4 Real Integrals: Some Special Contours.- 9.5 Infinite Series.- 10. Further Topics.- 10.1 Integration of f?/f; Rouché's 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 Möbius Transformations.- 11.4 Other Transformations.- 12. Final Remarks.- 12.1 Riemann's Zeta function.- 12.2 Complex Iteration.- 13. Solutions to Exercises.- Subject IndexBibliography.- Subject IndexIndex.

From the reviews: