Complex Made Simple

Ullrich, David

doi:10.1090/gsm/097

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Complex Made Simple

About this Title

David C. Ullrich, Oklahoma State University, Stillwater, OK

Publication: Graduate Studies in Mathematics
Publication Year: 2008; Volume 97
ISBNs: 978-0-8218-4479-3 (print); 978-1-4704-1163-3 (online)
DOI: https://doi.org/10.1090/gsm/097
MathSciNet review: MR2450873
MSC: Primary 30-01

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Chapter 0. Differentiability and the Cauchy-Riemann equations
Chapter 1. Power series
Chapter 2. Preliminary results on holomorphic functions
Chapter 3. Elementary results on holomorphic functions
Chapter 4. Logarithms, winding numbers and Cauchy’s theorem
Chapter 5. Counting zeroes and the open mapping theorem
Chapter 6. Euler’s formula for $\sin (z)$
Chapter 7. Inverses of holomorphic maps
Chapter 8. Conformal mappings
Chapter 9. Normal families and the Riemann mapping theorem
Chapter 10. Harmonic functions
Chapter 11. Simply connected open sets
Chapter 12. Runge’s theorem and the Mittag-Leffler theorem
Chapter 13. The Weierstrass factorization theorem
Chapter 14. Carathéodory’s theorem
Chapter 15. More on $\mathrm {Aut}(\mathbb {D})$
Chapter 16. Analytic continuation
Chapter 17. Orientation
Chapter 18. The modular function
Chapter 19. Preliminaries for the Picard theorems
Chapter 20. The Picard theorems

Part 2. Further results

Chapter 21. Abel’s theorem
Chapter 22. More on Brownian motion
Chapter 23. More on the maximum modulus theorem
Chapter 24. The Gamma function
Chapter 25. Universal covering spaces
Chapter 26. Cauchy’s theorem for nonholomorphic functions
Chapter 27. Harmonic conjugates

Part 3. Appendices

Appendix 1. Complex numbers
Appendix 2. Complex numbers, continued
Appendix 3. Sin, cos and exp
Appendix 4. Metric spaces
Appendix 5. Convexity
Appendix 6. Four counterexamples
Appendix 7. The Cauchy-Riemann equations revisited

Complex Made Simple

About this Title

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