C O M P L E X   A N A L Y S I S,   S P R I N G  2 0 2 3

Lectures: Tuesday and Thursday, 3.30pm-4.45pm, in Warren Weaver Hall 109.

Lecturer: Paul Bourgade, office hours Thursday 9.15-10.15am (Warren Weaver Hall 629), you also can email me (bourgade@cims.nyu.edu) to set up an appointment.

Recitations: By Eric Thoma (eric.thoma@cims.nyu.edu). He will run recitation sections in 194M 304 on Friday, 9.30am-10.45am and 3.30pm-4.45pm.

Course description: A one-semester introduction to complex analysis. The topics included are: complex numbers, differentiation of complex functions, Cauchy-Riemann equations, analytic functions, Cauchy's theorem and the Cauchy integral formula, Taylor series and Laurent series, singularities, residues, analytic continuation. The course will be mainly proof based, up to some calculus results taken as granted.

Textbook: The textbook for the course is Complex Variables and Applications , by James W. Brown and Ruel V. Churchill (9th edition, but previous editions are fine). Readings associated to each lecture are indicated on the calendar; you are encouraged to read these before coming to class. They contain in particular additional examples and exercises.

Grading: Your course score will be determined as the following weighted average:

Homework: 20%
Quizzes: 15%
Midterm 1: 20%
Midterm 2: 20%
Final Exam: 25% (+10% to replace part of the lowest midterm score if this score is better)

This score will be converted to a letter grade with the values below as cutoffs. These cutoffs might be adjusted at the end of the semester (the so-called ``curve''), but only in the downward direction (to make letter grades higher).

A : [100,93]
A- : (93,90]
B+ : (90,87]
B : (87,83]
B- : (83,80]
C+ : (80,75]
C : (75,65]
D : (65,50]
F : (50,0]

Quizzes: Quizzes will be available on NYU Brightspace, 48 hours before Tuesday class start time. They will include relatively basic questions concerning the lectures of the previous week. Students can select any 20 minute interval of convenience during this window to complete the quiz. Your response should be written individually without consulting resources other than the textbook and class notes. There will be no makeup quizzes (no exceptions). Two quiz scores will be dropped from your course grade.

Homework: HW will appear on this page each Thursday and be due the next Thursday, before 3.30pm either on paper or via email as a pdf file. They will include more advanced problems, mainly on the topics of the lectures of the previous week. Grading of homework will be based on clarity and correctness of mathematical arguments: you should detail the steps of your reasoning and cite theorems and definitions used. In fairness to graders and other students in the course, late homework will not be accepted (no exceptions). Two HW score will be dropped from your course grade.

Students are encouraged to work together, but submitted assignments must be written individually in your own words. Submitting two very similar sets of solutions is a violation of academic integrity and will be disciplined by the university. The best way to ensure this does not happen is to write your solutions separately.

Exams: There will be two midterm exams and one final exam. The midterm exams will be taken in class and last one hour. The final exam will be taken after the end of classes. During the exam, you will be allowed to consult the textbook and class notes (including exercises and HW solutions), but no other resources. Topics included are indicated below: this includes only the parts of the chapters covered in class.

A tentative schedule for this course is:

Day Date Textbook Topics Assignment
Tu 01/24 Chap 1: 1-4 Complex numbers
Th 01/26 Chap 1: 5-7 Conjugate and exponential form Quiz 1
Fr 01/27 No recitation
Tu 01/31 Chap 1: 8-11 Properties of the exponential and roots Quiz 2
Th 02/02 Chap 2: 13-15 Functions and limits HW 1+2
Fr 02/03 Recitation
Tu 02/07 Chap 2: 16-18 Limits and continuity Quiz 3
Th 02/09 Chap 2: 19-21 Differentiation and Cauchy-Riemann equations HW 3
Fr 02/10 Recitation
Tu 02/14 Chap 2: 22-25 More on C-R equations and analytic functions Quiz 4
Th 02/16 Chap 3: 30-31 The exponential and logarithm functions HW 4
Fr 02/17 Recitation
Tu 02/21 Chap 3: 32-34 More on logarithmic functions Quiz 5
Th 02/23 Chap 3: 35-38 The power and trigonometric functions HW 5
Fr 02/24 Recitation
Tu 02/28 Chapters 1 to 3 Midterm 1
Th 03/02 Chap 4: 41-43 Integral along real lines, contours
Fr 03/03 Recitation
Tu 03/07 Chap 4: 43-45 Contour integrals Quiz 6
Th 03/09 Chap 4: 46-48 Upper bounds and antiderivatives HW 6
Fr 03/10 Recitation
Tu 03/21 Chap 4: 48-49 Antiderivatives Quiz 7
Th 03/23 Chap 4: 50-52 Cauchy-Goursat theorem and simply connected domains HW 7
Fr 02/24 Recitation
Tu 03/28 Chap 4: 53-54 Multiply connected domains and Cauchy integral formula Quiz 8
Th 03/30 Chap 4: 55-56 General Cauchy integral formula HW 8
Fr 03/31 Recitation
Tu 04/04 Chapters 1 to 4 Midterm 2
Th 04/06 Chap 4: 57-58 Consequences of Cauchy integral formula
Fr 04/07 Recitation
Tu 04/11 Chap 5: 60-62 Sequences and series Quiz 9
Th 04/13 Chap 5: 63-64 Taylor series HW 9
Fr 04/14 Recitation
Tu 04/18 Chap 5: 65-68 Laurent series Quiz 10
Th 04/20 Chap 5: 69-71 Power series HW 10
Fr 04/21 Recitation
Tu 04/25 82,72,28 Uniqueness of series expansions and of analytic continuation Quiz 11
Th 04/27 Chap 6: 74-76 Isolated singularities and Cauchy’s residue theorem HW 11
Fr 04/28 Recitation
Tu 05/02 Chap 6: 77-79 Residue at infinity and classifications of singularities Quiz 12
Th 05/04 Chap 6: 80-81 Residues at Poles HW 12
Fr 05/05 Recitation

Homework.

Acknowledgement: The teaching material for this course was created by the excellent Michel Pain.