Upon completion of the course the student should be able to use various interpretations of complex numbers to solve algebraic and/or geometric problems; understand the notions of holomorphic and analytic functions and their central role in the theory of complex functions; differentiate and integrate complex functions; prove and apply Cauchy's theorem; rigorously work with power series; apply the residue theorem to calculate real integrals.

• Classes: MWF 8:00am-8:50am, NDSU Minard Hall Rm 308
• Office hours: MWF 12:00pm-12:50pm or by appointment (Minard 408E22)
• Syllabus: (pdf)
• Textbook: A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka, version 1.54, free pdf can be found at link. You can also buy an inexpensive printed version at Amazon.

• Visual Complex Analysis by Tristan Needham
• Complex Analysis (Princeton Lectures in Analysis, No. 2) by Elias M. Stein, Rami Shakarchi
• Complex Analysis (Undergraduate Texts in Mathematics) by Theodore Gamelin
• Complex Analysis by Lars Ahlfors
• Complex Made Simple (Graduate Studies in Mathematics) by David C. Ullrich

• The field of complex numbers
• Differentiation
• Integration
• Cauchy's theorem
• Harmonic functions
• Power series
• Taylor and Laurent series
• Residue theorem

• 1. How it all began with the complex numbers
• 2. Rigorous definition and geometric interpretation
• 3. More geometry of the complex plane
• 4. Functions of the complex variable
• 5. Cauchy--Riemann equations. Geometric meaning of derivative
• 6. A zoo of holomorphic functions
• 7. Complex log and multivalued functions (coming soon)
• 8. Mobius transformation (see the textbook)
• 9. Complex integral
• 10. Cauchy's theorem
• 11. Cauchy's integral and first applications
• 12. Fundamental theorem of algebra
• 13. Functional series. Uniform convergence
• 14. Power series
• 15. Analytic functions
• 16. Laplace equation and harmonic functions