Complex Analysis, Fall 2018

Mon 12/3: A glimpse at Riemann surfaces.

Fri 11/30: Riemann mapping theorem.

Reading: Chapter 8.3.

Wed 11/28: Riemann mapping theorem.

Reading: Chapter 8.3, through Montel’s Theorem

Mon 11/26: Conformal mappings continued.

Reading: Rest of Chapter 8.2

Wed 11/21: Conformal mappings continued.

Reading: Chapter 8.2
Homework (Due Fri Nov 30): (8.5) #1-6, 12, 14.

Mon 11/19: Conformal mappings continued.

Reading: Chapter 8.1.1, 8.1.2, 8.2
Homework (Due date TBD): (8.5) #1-6.

Fri 11/16: Conformal mappings.

Reading: Chapter 8.1

Wed 11/14: Analytic continuation of Zeta.

Reading: Chapter 6.2, through Theorem 2.4.

Fri 11/9: Fourier transform and Poisson summation continued.

Reading: 4.1, 4.2

Wed 11/7: Basel problem, a bit on Fourier transform and Poisson summation.

Reading: 4.1, 4.2

Mon 11/5: Gamma function continued.

Reading: 6.1.2.

Fri 11/2: Weierstrass products, Hadamard’s theorem.

Reading: 5.4 continued, part of 5.5. I will state Hadamard’s theorem and give a partial proof, but not the full proof.
Homework problems (Due Friday November 9, along with previous problems): (5.6) #9, 11, 12, 15

Wed 10/31: Infinite products continued, Weierstrass products.

Reading: 5.3, 5.4 continued

Mon 10/29: Infinite products continued, Weierstrass products.

Reading: 5.3, 5.4.
Homework problems (due date TBD, sometime next week, with some others to be added): (5.6): #2, 6, 10.

Fri 10/26: Order of growth of entire functions, infinite products.

Reading: 5.2, 5.3.1

Wed 10/24: No lecture.

Mon 10/22: Jensen’s Formula.

Reading: 5.1

Fri 10/12: The Gamma function.

Reading: 3.1, 3.1.1
Homework: Supplemental homework problems, due Friday 10/26.

Wed 10/10: Complex logarithm continued.

Reading: 3.6.

Mon 10/8: Open mapping, simply connected domains, and the complex logarithm.

Reading: 3.4, 3.5, 3.6.

Wed 10/3: Argument principle.

Reading: 3.4.

Wed 10/3: Characterization of singularities, argument principle.

Reading: 3.3, 3.4.
Problems: No new homework problems. Oral exam problems will be distributed in class.

Mon 10/1: Residue Theorem Applications.

Reading: 3.2, maybe the start of 3.3
Problems: No new homework problems. Oral exam problems will be distributed Wednesday.

Fri 9/28: Zeroes and Poles and the Residue Theorem.

Reading: 3.1, 3.2.
Problems: (Due Wednesday 10/3) (3.8) #1, 2, 3.

Wed 9/26: Consequences of Cauchy’s integral formulas.

Reading: 2.5.1, 2.5.2, if time we will start 3.1
Problems: (no new problems)

Mon 9/24: Consequences of Cauchy’s integral formulas.

Reading: 2.4
Problems (Due Friday 9/28 along with previous problems): (2.6) #9.

Fri 9/14: Cauchy’s Integral Formulas, continued.

Reading: 2.2, 2.3
Problems (Due Friday 9/28 along with previous problems): (2.6) #8, 11, 12.

Wed 9/12: Cauchy’s Integral Formulas.

Reading: 2.2, 2.3
Problems: (2.6) #1, 2.

Mon 9/10: Cauchy-Goursat Theorem.

Reading: 2.1, 2.2
Problems: (2.6) #5, 6.

Fri 9/7: Integration along curves continued.

Reading: 1.2.3
Problems: No new problems this lecture. If you need additional time with the integration problems assigned last time (#24 and 25), you may turn those in next week.

Wed 9/5: Power series continued, contour integrals.

Reading: 1.2.3
Problems (due Friday 9/7 along with previous problems): Ch. 1: #13, 24, 25.

Fri 8/31: Holomorphic functions, power series.

Reading: 1.2.2
Problems (due Friday 9/7 along with previous problems): Ch. 1: #14, 18, 19, 23

Wed 8/29: Complex functions.

Reading: 1.2.1, 1.2.2
Problems (due date TBD): Ch. 1: #8, 10, 11, 12, 13

Mon 8/27: The Complex Plane.

Reading: 1.1.1-1.1.2
Problems (due date TBD): Ch. 1: #1-4, 6, 7.