This volume studies functions of a complex variable. It develops analyticity, contour integration, and the rich structure that arises from complex differentiability.
Part I. Complex Numbers and Functions
Chapter 1. Complex Numbers
1.1 Algebra of complex numbers 1.2 Geometry in the complex plane 1.3 Polar form and Euler’s formula 1.4 Roots of unity 1.5 Basic properties
Chapter 2. Complex Functions
2.1 Functions of a complex variable 2.2 Limits and continuity 2.3 Differentiability 2.4 Holomorphic functions 2.5 Examples
Chapter 3. Analyticity
3.1 Power series 3.2 Analytic functions 3.3 Cauchy–Riemann equations 3.4 Harmonic functions 3.5 Examples
Part II. Complex Integration
Chapter 4. Contours and Integrals
4.1 Paths in the complex plane 4.2 Line integrals 4.3 Properties 4.4 Examples 4.5 Applications
Chapter 5. Cauchy Theory
5.1 Cauchy integral theorem 5.2 Cauchy integral formula 5.3 Consequences 5.4 Applications 5.5 Examples
Chapter 6. Residue Theory
6.1 Singularities 6.2 Residues 6.3 Residue theorem 6.4 Evaluation of integrals 6.5 Applications
Part III. Series and Expansions
Chapter 7. Power Series
7.1 Convergence 7.2 Radius of convergence 7.3 Operations on series 7.4 Examples 7.5 Applications
Chapter 8. Laurent Series
8.1 Expansion near singularities 8.2 Principal parts 8.3 Classification of singularities 8.4 Examples 8.5 Applications
Chapter 9. Analytic Continuation
9.1 Extension of functions 9.2 Monodromy (overview) 9.3 Branch points 9.4 Riemann surfaces (overview) 9.5 Applications
Part IV. Conformal Mapping
Chapter 10. Conformal Maps
10.1 Angle preservation 10.2 Basic examples 10.3 Möbius transformations 10.4 Properties 10.5 Applications
Chapter 11. Mapping Theorems
11.1 Riemann mapping theorem (overview) 11.2 Schwarz lemma 11.3 Automorphisms of domains 11.4 Applications 11.5 Examples
Chapter 12. Applications
12.1 Fluid flow 12.2 Electrostatics 12.3 Boundary value problems 12.4 Engineering applications 12.5 Examples
Part V. Special Functions and Advanced Topics
Chapter 13. Entire and Meromorphic Functions
13.1 Definitions 13.2 Growth of functions 13.3 Zeros and poles 13.4 Weierstrass factorization (overview) 13.5 Applications
Chapter 14. Special Functions
14.1 Gamma function 14.2 Zeta function 14.3 Elliptic functions (overview) 14.4 Functional equations 14.5 Applications
Chapter 15. Value Distribution Theory (Overview)
15.1 Nevanlinna theory basics 15.2 Distribution of values 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Harmonic and Potential Connections
Chapter 16. Harmonic Functions
16.1 Laplace equation 16.2 Mean value property 16.3 Maximum principle 16.4 Applications 16.5 Examples
Chapter 17. Potential Theory Links
17.1 Green’s functions 17.2 Dirichlet problem 17.3 Boundary behavior 17.4 Applications 17.5 Examples
Chapter 18. Fourier and Complex Methods
18.1 Fourier series via complex analysis 18.2 Integral transforms 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Computational and Applied Aspects
Chapter 19. Numerical Complex Analysis
19.1 Approximation methods 19.2 Contour integration techniques 19.3 Stability issues 19.4 Applications 19.5 Examples
Chapter 20. Applications
20.1 Physics 20.2 Engineering 20.3 Signal processing 20.4 Number theory links 20.5 Examples
Chapter 21. Computational Tools
21.1 Symbolic computation 21.2 Numerical libraries 21.3 Visualization 21.4 Software systems 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Several complex variables (overview) 22.2 Complex dynamics 22.3 Teichmüller theory (overview) 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Growth and distribution questions 23.2 Mapping problems 23.3 Functional equations 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of complex analysis 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common identities and formulas B. Residue computation table C. Proof techniques checklist D. Example catalog E. Cross-reference to other MSC branches
This volume develops complex analysis as a highly structured theory of functions. It emphasizes analyticity, integration, and geometric properties unique to the complex domain.