Please spend at least 2 hours a night reading the material/looking at the proofs/making sure you understand the details. Below is a tentative reading list and homework assignments. It is subject to changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.

• Week 1: September 6 to 10
  • Read: Chapter 1 and my online notes, review your real analysis. Useful resources are the online real analysis book
  • HW: Due by 10am Friday, September 17: Chapter 1: Page 24: #1abcd, #3, #13.
  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 1: #5, #7, #16d, #18, #23, #26.
  • Extra credit: see the extra credit list.

• Week 2: September 13 to 17
  • Read: Chapter 1 and my online notes, review your real analysis. Useful resources are the online real analysis book
  • Read Chapter 2 and my online notes for Sections 2.1, 2.2 and 2.3.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due in my mailbox by 10am Friday, September 24: Chapter 1: Page 24: #16abc, #24, #25ab. Chapter 2: (#1) We proved Goursat's theorem for triangles. Assume instead we know it holds for any rectangle; prove it holds for any triangle. (#2) Let gamma be the closed curve that is the unit circle centered at the origin, oriented counter-clockwise. Compute Integral_gamma f(z) dz where f(z) is complex conjugation (so f(x+iy) = x - iy). Repeat the problem for Integral_gamma f(z)^n dz for any integer n (positive or negative), and compare this answer to the results for Integral_gamma z^n dz; is your answer surprising? (#3) Prove that the four triangles in the subdivision in the proof of Goursat's theorem are all similar to the original triangle. (#4) In the proof of Goursat's theorem we assumed that f was complex differentiable (ie, holomorphic). Would the result still hold if we only assumed f was continuous? If not, where does our proof break down?
  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 1: #5, #7, #16d, #18, #23, #26.

   

• Week 3: September 20 to 24
  • Read Chapter 2 and my online notes for Section 2.4.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due in my mailbox by 10am Friday, October 1 (even if this is Mountain Day): Chapter 2, Page 64: #1, #8. Also do: Chapter 2: (#1) In the proof of Liouville's theorem we assumed f was bounded. Is it possible to remove that assumption? In other words, is it enough to assume that |f(z)| < g(z) for some real-valued, non-decreasing function g? If yes, how fast can we let f grow?  (#2) a) Find all z where the function f(z) = 1 / (1+z⁴) is not holomorphic; b) Let a, b, c, and d be integers such that ad - bc = 1. Find all z where the function g(z) = (az + b) / (cz + d) is not holomorphic. (#3) Compute the power series expansion of f(z) = 1 / (1-z) about the point z = 1/2 (it might help to do the next problem first, or to write 1 - z as 1/2 - (z - 1/2)). (#4) Do Chapter 1, Page 29, #18.
  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 2: #2, #3, #5, #6. Also evaluate the integral of sin²x/x² for -∞ < x < ∞. Also, the fundamental theorem of algebra says a polynomial of degree n has n complex roots. The roots thus lie in a disk of radius R. Find an upper bound for R in terms of the coefficients of the polynomial and its degree; in other words, find a computable number R (as a function of the parameters of the problem) such that all the roots must lie in the disk |z| < R. Finally, read about a fundamental theorem you don’t already know.

   

• Week 4: September 27 to October 1
  • Read Chapter 3 (Sections 3.1, 3.2, 3.3 and 3.4) and my online notes.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due in my mailbox by 10am Friday, October 8 (even if this is Mountain Day): Chapter 3, Page 103: #1, #2, #5 (this is related to the Fourier transform of the Cauchy density), #15d, #17a (hard). Additional: Let f(z) = Sum_{n = -5 to oo} a_n zⁿ and g(z) = Sum_{m = -2 to oo} b_m z^m be the Laurent expansions for two functions holomorphic everywhere except possibly at z=0. a) Find the residues of f(z) and g(z) at z=0; b) Find the residue of f(z)+ g(z) at z=0; c) Find the residue of f(z) g(z) at z=0; d) Find the residue of f(z)/g(z) at z=0.
  • Suggested Problems: Chapter 3: Chapter 3: Do #2 with x^2k instead of x⁴, #3, #9, #12 (great problem: also take u = 1/2 and deduce a formula for the sum of 1/n² -- amazingly, this should be doable from just knowing the sum over the odd integers!), #13, #15c, #19. Also: is there a nice formula for the residue of f(g(z)), where f and g are the functions from the Additional problem above?

   

• Week 5: October 4 to 8
  • Read Chapter 3 (except for Section 5) and my online notes. Just skim Section 3.3.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due in my mailbox by 10am Friday, October 22 (as there is no class on Tuesday): Chapter 5: Page 155: #6, #7, #9 (extra credit: what is the combinatorial significance of this problem?). Chapter 3: Page 104: #10. Additional Problems: (1) Find all poles of the function f(z) = 1 / (1 - z²)⁴ and find the residues at the poles. (2) Consider the map f(z) = (z - i) / (z + i). Show that this is a 1-to-1 and onto map from the upper half plane (all z = x + iy with y > 0) to the unit disk. (3) Calculate the Weierstrass product for cos(πz) (this is also problem #10b in Chapter 5, and the answer is listed there) and for tan(πz).
  • Suggested Problems: Chapter 5: #1, #10a, #13 (you'll have to read about Hadamard's theorem).

• Week 6: October 11 to 15: No class on Tuesday
  • Read Chapter 5 (Introduction, Section 3 and Section 4) and my online notes.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due in my mailbox by 10am Friday, October 22 (as there is no class on Tuesday, October 12): Chapter 5: Page 155: #6, #7, #9 (extra credit: what is the combinatorial significance of this problem?). Chapter 3: Page 104: #10. Additional Problems: (1) Find all poles of the function f(z) = 1 / (1 - z²)⁴ and find the residues at the poles. (2) Consider the map f(z) = (z - i) / (z + i). Show that this is a 1-to-1 and onto map from the upper half plane (all z = x + iy with y > 0) to the unit disk. (3) Calculate the Weierstrass product for cos(πz) (this is also problem #10b in Chapter 5, and the answer is listed there) and for tan(πz). In honor of the good attendance for Mountain Day, one problem is `optional' and thus the maximum score is 70/60.
  • Suggested Problems: Chapter 5: #1, #10a, #13 (you'll have to read about Hadamard's theorem).

   

• Week 7: October 18 to 22: Take home midterm the following week; Green Chicken 10:30am in Bronfman 106
  • Read Chapter 5, Section 4 and  my online notes; read Chapter 3 (the proof of the Open Mapping Theorem in Section 4, and Section 6) and online notes.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • No HW as there is a take-home midterm, though you should look at the HW due November 5th and start it as soon as you can!
  • Extra Credit: Due Friday Nov 5: Find a sequence a_n of real numbers that is conditionally convergent but not absolutely convergent such that the product Prod_n (1 + a_n) diverges, or show that no such series exists.

   

• Week 8: October 25 to 29: Take home midterm the following week; Green Chicken 10:30am in Bronfman 106
  • Read Chapter 8 and my online notes.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due Friday, November 5: (1) Evaluate Integral_{-oo to oo} cos(4x)dx / (x^4 + 1). (2) Let U be conformally equivalent to V and V conformally equivalent to W with functions f: U --> V and g: V --> U. Prove g ◦ f (g composed with f) is a bijection. (3) The Riemann mapping theorem asserts that if U and V are simply connected proper open subsets of the complex plane then they are conformally equivalent. Show that simply connected is essential. In other words, find a bounded open set U that is not simply connected and prove that it cannot be conformally equivalent to the unit disk. (4) Chapter 8, Page 248: #4. (5) Chapter 8: Page 248: #5. (6) Chapter 8: Page 251: #14.
  • Suggested Problems: TBD
  • Extra Credit: Due Friday Nov 5: Find a sequence a_n of real numbers that is conditionally convergent but not absolutely convergent such that the product Prod_n (1 + a_n) diverges, or show that no such series exists.

   

• Week 9: November 1 to 5:
  • Read Chapter 8 (Sections 1, 2 and 3) and my online notes. We won't do harmonic functions or Dirichlet's problem.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • Handout on uniform continuity and some of the analysis material
  • HW: Due Friday, November 5: (1) Evaluate Integral_{-oo to oo} cos(4x)dx / (x^4 + 1). (2) Let U be conformally equivalent to V and V conformally equivalent to W with functions f: U --> V and g: V --> W. Prove g ◦ f (g composed with f) is a bijection. (3) The Riemann mapping theorem asserts that if U and V are simply connected proper open subsets of the complex plane then they are conformally equivalent. Show that simply connected is essential. In other words, find a bounded open set U that is not simply connected and prove that it cannot be conformally equivalent to the unit disk. (4) Chapter 8, Page 248: #4. (5) Chapter 8: Page 248: #5. (6) Chapter 8: Page 251: #14.
  • Extra Credit: Due Friday Nov 5: Find a sequence a_n of real numbers that is conditionally convergent but not absolutely convergent such that the product Prod_n (1 + a_n) diverges, or show that no such series exists.
  • Extra Credit: Due Friday Nov 5: Let f: (-1,1) to (-1,1) be a real analytic automorphism. What is the largest f'(0) can be? We showed in class that it could be as large as pi/2, and then as large as 2. Can you do better?

   

• Week 10: November 8 to 14:
  • Read Chapter 8 (Sections 1, 2 and 3) and my online notes. We won't do harmonic functions or Dirichlet's problem.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • Handout on uniform continuity and some of the analysis material
  • HW: Due Friday, November 12: DO ANY FIVE OUT OF THE FOLLOWING SIX: IF YOU DO MORE, THAT'S GOOD BUT ONLY THE FIRST FIVE WILL BE GRADED. (1) Consider the functions f_n(x) = n / (1 + n x^2) where n is a positive integer. Prove that each f_n is uniformly continuous on the real line. Is the family {f_n: n a positive integer} equicontinuous on compact sets? (2) Consider a 2x2 matrix M with integer entries and top row (a,b) and bottom row (c,d) such that ad-bc = 1; we denote the set of all such matrices by SL(2,Z). Consider the map f_M(z) = (az + b)/(cz+d) with z in the upper half plane. Is the family {f_M: M in SL(2,Z)} uniformly bounded on compact sets of the upper half plane? Hint: I think each map is bounded on compact subsets of the upper half plane, but you can find a sequence of matrices such that no bound works simultaneously. (3) Let f_n(x) = 1 - nx for 0 <= x <= 1/n and 0 otherwise, and let F = {f_n: n a positive integer}. Prove that lim f_n exists and determine it. (4) Consider the family from (3). Prove it is not normal (the problem is that the convergence is not uniform). Specifically, to be normal not only must it converge, but given any epsilon there is an N such that, for all n > N, |f_n(x) - f(x)| < epsilon (or this must hold for a subsequence). (5) Evaluate the integral from -oo to oo of x^2 / (x^4 + x^2 + 1). (6) Integrate from 0 to 2pi the function 1 / (a + b sin theta) where a and b are real numbers. What restrictions must we place on a and b in order for this to make sense?
  • Extra Credit: Due Friday Nov 12: Let F be the family of functions {cos(2 pi sqrt(n) + x): n a positive integer}. Determine whether or not this family is normal (which means that given any sequence of functions in F there is a subsequence that converges, not necessarily to something in F).

   

• Week 11: November 15 to 21:
  • Read: Contour Integration Revisited: Lecture notes from Tapper as well as Wapedia entry.
  • Read also: Lecture notes on examples of contour integrals
  • HW: Due Tuesday, November 23rd: DO ANY FIVE OUT OF THE FOLLOWING SIX: IF YOU DO MORE, THAT'S GOOD BUT ONLY THE FIRST FIVE WILL BE GRADED. (1) Let Omega be the subset of the complex plane of all z = x+iy with |x| < |y|. Does there exist a logarithm on Omega? If yes, what does the image of Omega under the logarithm look like? (2) Let Omega be the region from (4); is Omega conformally equivalent to the unit disk? Prove your assertion. Hint: remember the full definition of what it means to be simply connected. (3) Let Omega be the subset of the complex plane of all z = x + iy with |x| < |y| + 1. Conformally map Omega onto an open subset of the disk -- you must give an explicit form for the map. (Note: The Riemann Mapping Theorem asserts that you can get a map that is onto the disk; here you are just being asked to get a map that is holomorphic and 1-1). (4) Evaluate Int_{0 to 2 Pi} Cos[x]^m dx, where m is a positive integer. (5) Evaluate Int_{0 to 1} (1 - x^2)^n dx, where n is a positive integer. Hint: Let x = Sin[theta], dx = Cos[theta] d theta. (6) Int_{0 to oo} log(x) dx / (1 + x^2).
  • Consider a look-up table for cosine, where we record the value of cos(theta_n) for theta_n = 2 pi n / N, n = 0, 1, ..., N-1. For now, consider the following look-up procedure. For a given theta, find the largest n such that theta_n <= theta. What is the average error? In other words, what is (1/2pi) Int_{0 to 2pi} (cos theta - cos theta_n)^2 d theta as a function of N? I have a very nice expression for this. More generally, what if we take the closer of the two entries in our look-up table, or, even better, if we interpolate?