# Math 326 — Functions of a Complex Variable

Assignments are due on Thursday, at the beginning of class. The row that an assignment appears in is the day that it is due.

Date Topic Book Homework Practice Problems
Sept. 13 Introduction to the course §1.1–1.2   1.1: 8, 10; 1.2: 7, 10
15 More about the complex numbers §1.3   8, 11, 12
16 Visualizing complex mappings §1.3
20 Möbius transformations §1.4–1.5 1.4: 5, 7, 11; 1.5: 6, 12, 18
22 Exponentials of complex numbers §3.3, 3.5 3.3: 1, 5; 3.5: 1, 4
23 Limits and continuity §2.1–2.2 H1 2.1: 1, 6; 2.2: 6, 17
27 Holomorphicity §2.3 1, 4, 6, 10
29 The Cauchy-Riemann equations §2.4 1, 2, 3, 4
30 Proof of the Cauchy-Riemann Theorem §2.4 H2 3, 5, 15

Oct. 4 Harmonic Functions §2.5 1, 2, 3, 5
6 Differentiation of elementary functions §3.1–3.3 3.2: 8, 9, 10; 3.3: 6
7 Contour integrals §4.1–4.2 H3 4.1: 5, 8; 4.2: 6, 12
11 Thanksgiving
13 Properties of contour integrals §4.2–4.3 4.3: 2, 4, 5, 7
14 Fundamental theorem of complex calculus §4.3 H4
18 Cauchy's theorem §4.4 1, 3, 4, 10, 11, 18
20 More about Cauchy's theorem §4.4
21 Proof of Cauchy's theorem §4.4 H5
25 Cauchy's integral formula §4.5 1, 2, 8, 13
27 Some consequences of the integral formula §4.6 1, 3, 5, 6,
28 Maximum modulus theorem and harmonic functions §4.6–4.7 H6 4.6: 8, 14; 4.7: 4, 6

Nov. 1 The Dirichlet problem and Poisson's formula §4.7 8, 9, 10, 11
3 Convergent series of analytic functions §5.1, 5.4 5.1: 10, 11; 5.4: 5, 8
4 Taylor series expansions of holomorphic functions §5.2 H7 5.2: 1, 2, 11
8 Some corollaries of Taylor series expansions §5.3 1, 4, 5, 10
10 Laurent series §5.5 1, 2, 6, 7
11 Classification of singularities §5.6 H8 1, 2, 5, 6
15 Residue theorem and calculation of residues §6.1 1, 2
17 More residue calculations §6.1 5, 6, 7
18 Definite integrals I §6.3 H9 1, 3, 4, 5
22 Definite integrals II §6.5 2, 3, 4, 5
24 Definite integrals III §6.4 1, 3, 4, 6
25 Definite integrals IV §6.2 H10 1, 4, 6

29 The principle of the argument §6.7 1, 2, 3, 4
Dec. 1 Rouché's theorem §6.7 6, 7, 8, 9
2 Review   H11
6
8
13   H12