MAT 527 -- Complex Variables I 
Course Information

 
Professor: D. Bradley

Office: 322 Neville Hall 
Hours: 11:00-11:50 a.m. MWF^* 
Phone: (207) 581-3920

Grades:

Grades will be assigned on the basis of in-class participation and successful completion of written homework assignments.

Text:

Reinhold Remmert, Theory of Complex Functions (English ed. translated by Robert B. Burckel), Springer-Verlag Graduate Texts in Mathematics #122, New York, 1991.

Syllabus: I plan to cover most of chapters 1 through 8. In slightly more detail:

• Part A. Elements of Function Theory
• Chapter 0. Complex Numbers and Continuous Functions
• The field C of complex numbers
• Topological Background
• Chapter 1. Complex-Differential Calculus
• Distinction between complex-differentiable and real-differentiable functions
• Cauchy-Riemann equations
• Holomorphic functions
• Chapter 2. Holomorphy and Conformality
• Angle-preserving mappings
• Biholomorphic mappings
• Automorphisms of the upper half-plane and the unit disc
• Chapter 3. Modes of Convergence
• Chapter 4. Power Series
• Convergence criteria
• Exponential, trigonometric, logarithmic and binomial series
• Holomorphy of power series
• Algebra of convergent power series
• Chapter 5. Elementary Transcendental functions
• The exponential and trigonometric functions
• The epimorphism theorem for exp
• Polar coordinates, roots of unity and natural boundaries
• Logarithms
• Part B. Cauchy Theory
• Chapter 6. Complex Integral Calculus
• Integration over real intervals
• Path integrals in C
• Properties of complex path integrals
• Path independence of integrals
• Chapter 7. Cauchy's Integral Formula
• Goursat's lemma
• Cauchy's integral formula for discs
• Power series expansions of holomorphic functions
• Analytic continuation, radii of convergence
• Bernoulli numbers
• Part C. Cauchy-Weierstrass-Riemann Function Theory
• Chapter 8. Fundamental Theorems about Holomorphic Functions
1. The identity theorem
2. Holomorphy
3. Cauchy's inequalities for Taylor coefficients
4. Weierstrass's convergence theorem
5. Open mapping theorem and maximum principle
• Additional topics as time permits. MAT 528 will continue where we leave off.