Complex analysis can be a difficult subject and many introductory texts
are just too ambitious for today's students. This book takes a lower
starting point than is traditional and concentrates on explaining the key
ideas through worked examples and informal explanations, rather than
through "dry" theory, making it suitable for both pure and applied
mathematicians.

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.- SubjectIndexBibliography.- Subject IndexIndex.