Schedule


Lecture Date Topic Reading* Notes
1 Jan 21 Complex numbers 1-6
2 Jan 23 Euler's formula 7-12 HW 1 due
3 Jan 26 Functions 13-14
4 Jan 28 Limits 15-16
5 Jan 30 Continuity 17-18 HW 2 due
6 Feb 2 Differentiation: The Cauchy-Riemann equations 19-21
7 Feb 4 Sufficient conditions for differentiability 22-23
8 Feb 6 Analytic functions and polar coordinates 24-26 HW 3 due
9 Feb 9 Exponentials and logarithms 30-34
10 Feb 11 Power and trigonometric functions 35-38
Feb 13 Midterm I
Feb 16 Holiday (No class)
11 Feb 18 Contour Integrals 40-44
12 Feb 20 Examples and path independence 45-49 HW 4 due
13 Feb 23 Cauchy-Goursat Theorem 50-51
14 Feb 25 Cauchy-Goursat and Antiderivatives 52-53
15 Feb 27 Cauchy's Integral Formula 54-56 HW 5 due
16 Mar 2 Cauchy's Integral Formula and Harmonic functions 57,27
17 Mar 4 Liouville's Theorem, maximum modulus principle 57-59
18 Mar 6 Sequences and Series 60-61 HW 6 due
19 Mar 9 Taylor Series 62-64
20 Mar 11 Laurent Series 65-67
21 Mar 13 Some examples, uniform convergence of power series 68-69 HW 7 due
22 Mar 16 Integration, differentiation of series, etc. 70-73
Mar 18 Cauchy-Residue Theorem 74-77
Mar 20 Midterm II
Mar 23 Spring Recess (No class)
Mar 25 Spring Recess (No class)
Mar 27 Spring Recess (No class)
23 Mar 30 Types of singularities, residues at poles 78-80
24 Apr 1 Zeros of analytic functions 81-82
25 Apr 3 Poles and singularities 83-84 HW 8 due
26 Apr 6 Applications to improper integrals 85-87
27 Apr 8 Jordan's Lemma and indented paths 88-89
28 Apr 10 Branch cuts and trigonometric integrals 90-92 HW 9 due
29 Apr 13 Argument principle and Rouche's theorem 93-94
30 Apr 15 Linear transformations, the mapping w=1/z 96-98
31 Apr 17 Fractional linear transformations 99-102 HW 10 due
32 Apr 20 The mappings exp(z) and sin(z) 103-106
33 Apr 22 Mappings by square roots and polynomials 107-109
34 Apr 24 Riemann surfaces 110-111, G-XVI.1** HW 11 due
35 Apr 27 Conformal mappings and inverse function theorem 112-114
36 Apr 29 Analytic continuation and the Gamma function G-XIV.1**
37 May 1 The Riemann-Zeta function G-XIV.3** HW 12 due
May 6 Final exam review I
May 8 Final exam review I
May 12 Final Exam: 7pm-10pm

*The numbers in the reading column refer to chapters in the course textbook
Brown and Churchill. Complex variables and Applications, 9th Edition, 2013.

**Numbers of the form G-# refer to chapters in the book
Theodore W. Gamelin, Complex Analysis, Springer, 2001,
which is available on SpringerLink.