Prof. Jim Haglund, jhaglund@math.upenn.edu
Course webpage: http://www.math.upenn.edu/~jhaglund/4100/

Office hours for Dec. 7-20: F Dec. 8 10am-11am, M Dec. 11 10:30am-11am and 11:30am-noon, W Dec. 13 8:30-9:30am, T Dec. 19, 10am-11am, W Dec. 20, 10:00am-11:00am in DRL 4e2b.

Office Phone: (215) 573-9093.

Grader: Nikita Borisov
nborisov@sas.upenn.edu

Grader's Office hours: Tuesday 10:30am-11:30am in DRL 3C9.

Lecture: TR 12 noon - 1:29pm in Towne 313

Course: An introduction to the field of Complex Analysis, suitable for math majors and advanced undergraduate or beginning graduate students from engineering, physics, or applied mathematics. The primary text for the course is Complex Analysis by Bak and Newman, third edition, 2010, which should be available soon at the Penn bookstore.(The material we will cover is fairly standard, and any other book in Complex Analysis, such as "Complex Analysis and its Applications" by Churchill, will do.)
Specific topics to be covered in the course include:

Basic properties of complex numbers and the geometry of the complex plane. Roots of unity.

Analytic functions and the Cauchy-Riemann equations. Power series and differentiability. Complex version of the exponential, logarithm, sine and cosine functions.

Maximum modulus theorem, Schwarz's lemma, Laurent expansions.

Integration, winding numbers and Cauchy's residue theorem, contour integration.

Elementary conformal mapping.

Exams and Grades: There will be two hour exams and a final exam. Each hour exam counts 25% of your grade. The final exam has two components; an in-class exam counting 25% of your grade and a take-home exam worth 15% of your grade. There will also be HW assignments (posted at the bottom of this web page) counting 10% of your grade. Midterm 1 will be on Thursday, Oct. 5 from 12-1:29pm in Towne 313 and Midterm 2 will be on Thursday, Nov. 16 from 12-1:29pm, also in Towne 313. Both the midterms and the in-class final exam will be closed-book: no calculators, smart phones, etc.

Midterm 1: The first midterm is a closed book exam. No books, notes, smart phones, calculators, etc. are allowed. It covers all the material from Chapters 1-5 of Bak and Newman's "Complex Analysis", third edition, except section 5.3 which we will be skipping. (The exam also includes material from lecture on roots of unity.)

Midterm 2: The second midterm is a closed book exam. No books, notes, smart phones, calculators, etc. are allowed. It covers all the material from Chapters 6-9 of Bak and Newman's "Complex Analysis", third edition. On the exam you will be asked to prove one or two of the four theorems below from the book (in addition to other problems; there will be a total of 6 problems).

Casorati-Weierstrass Theorem
The Schwartz Reflection Principle
Maximum Modulus Theorem
Cauchy Integral Formula (for functions analytic in a disk, as in Chapter 6)

Final Exam: There will be both an in-person exam and a take-home exam component to the final. The in-person component will count 25% of your total course grade, and the take-home component 15% of your total course grade. The in-person exam will be given on Thursday, Dec. 14 from 9am-11am in McNeil 150. It is a closed book exam. No books, notes, smart phones, calculators, etc. are allowed. It covers all the material from Chapters 1-9 of Bak and Newman's "Complex Analysis", third edition. On the exam you will be asked to prove one or two of the two theorems below from the book (in addition to other problems; there will be a total of 6 problems).

The Schwartz Reflection Principle
Maximum Modulus Theorem

The take-home exam is posted below. It consists of 6 problems, based on material from Chapters 10 and 11 of Bak and Newman on the Cauchy Residue Theorem. You will solve the problems, giving all important details, and upload a pdf containing your solutions to canvas. It s due on Dec. 20 at midnight.

Important Dates:
Tuesday, August 29: Classes begin
Midterm 1: Thursday, Oct. 5, 12-1:29pm in Towne 313
Last Day to Drop a Course: Oct. 9.
Fall Break: Thursday, Oct. 12 - Sunday, Oct. 15.
Last Day to Withdraw from a Course: Nov. 6.
Midterm 2: Thursday, Nov. 16, 12-1:29pm in Towne 313
Thanksgiving Break: Thursday, Nov. 23 - Sunday, Nov. 26
Last Day of Classes: Monday, Dec. 11
In Class Final exam: (Covering Chapters 1-9) Thursday, Dec. 14 from 9am-11am in McNeil 150
Take Home Final exam: (Covering Chapters 10 and 11) See link below. Due Wednesday, Dec. 20 at midnight.

Take-Home Final Exam

Homework Assignments:

HW1 (due in class on 9/14): 1) (#6 from p. 32); Prove that if f,g are differentiable at z, then so are f+g and fg, and (if g not equal to 0) f/g.
2) (#9, p. 32) Find the radius of convergence of the power series whose nth term is (a) z^(n!) and (b) (n+2^n)z^n
3) (#13 p. 33) (a) Suppose {a_n} is a sequence of positive reals and the limit as n -> infinity of (a_{n+1}/a_n) equals L. Show that the limit as n -> infinity of (a_n)^(1/n) = L.
(b) Use the result from (a) to prove the limit as n-> infinity of (1/n!)^(1/n) =0.
4) (#14 p. 33) Use part (a) above to find the radius of convergence of the series whose nth term is (a) (-1)^n z^n/n!
(b) z^(2n+1)/(2n+1)!
(c) n!z^n/(n^n)
(d) 2^n z^n/n!
5) (#20 p. 34) Prove Corollary 2.13 (that if a power series equals zero at all the points of a set with an accumulation point at the origin, the power series is identically zero) by showing that if a set S has an accumulation point at 0, it contains a sequence of nonzero terms which converge to 0.
6) (#10, p. 42) Suppose f is an entire function of the form f(x,y)=u(x) + iv(y). Show that f is a linear polynomial.

HW2.pdf (due in class on 9/28 or on canvas as a pdf upload)

HW3.pdf (due in class on 11/2 or on canvas as a pdf upload)

HW4.pdf (due in class on 11/14 or on canvas as a pdf upload)