MAT 528 -- Complex Variables II 
Course Information

 
Professor: D. Bradley

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

Time & Location: 10:00-10:50 a.m. MWF, 327 Neville Hall

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 9 through 14. In slightly more detail:

• Chapter 9. Miscellany
• § 3. Holomorphic logarithms and roots
• § 4. Local normal forms
• § 5. General Cauchy theory
• § 6. Asymptotic power series developments
• Chapter 10. Meromorphic Functions
• § 1. Isolated singularities, Casorati-Weierstrass theorem
• § 2. Automorphisms of punctured domains
• § 3. Meromorphic functions
• Chapter 11. Convergent Series of Meromorphic Functions
• § 1. General convergence theory
• § 2. The partial fraction expansion of pcotpz
• § 3. The Euler formulas for z(2n)
• § 4. Eisenstein's theory of the trigonometric functions
• Chapter 12. Laurent Series and Fourier Series
• § 1. Holomorphic functions in annuli
• § 2. Properties of Laurent series
• § 3. Periodic holomorphic functions and Fourier series
• § 4. The theta function
• Chapter 13. The Residue Calculus
• § 1. The residue theorem
• § 2. Consequences of the residue theorem: counting zeros and poles, Rouche's theorem
• Chapter 14. Definite Integrals
• § 1. Calculation of integrals
• § 2. Further evaluation of integrals
• § 3. Gauss sums
• Additional topics as time permits.