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**.

- Homework 1+2. Solution
- Homework 3. Solution
- Homework 4. Solution
- Homework 5. Solution
- Homework 6. Solution
- Homework 7. Solution
- Homework 8.

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