Complex Analysis (Spring 2023)

Mathematics 633: Complex Analysis (Spring 2023)

Tu Th 8:30 AM - 9:45 AM        Physics 205

Midterm: March 9th, 8:30 am to 9:45 am

Final: April 18th, 8:30 am to 9:45 am

Professor: Saman Habibi Esfahani

Email: Saman.HabibiEsfahani@duke.edu

Textbook: Complex Analysis 3rd Edition by Lars Ahlfors

Homework: There will be weekly homework due on Thursdays (11:59 pm). I will post the homework here (and on Gradescope) each Thursday, and you should upload your homework on the Gradescope page of the course. Late homeworks will not be accepted. Typing in Latex is recommended.

Grader: Michael Lin

Homework 1, Solutions (due date: Jan 19)

Homework 2, Solutions (due date: Jan 26)

Homework 3, Solutions (due date: Feb 02)

Homework 4, Solutions (due date: Feb 09)

Homework 5, Solutions (due date: Feb 16)

Homework 6, Solutions (due date: Feb 23)

Homework 7, Solutions (due date: Mar 02)

Homework 8 (Practice Exam) (due date: Mar 09)

Midterm, Solutions (date: Mar 09)

Homework 9, Solutions (due date: Mar 23)

Homework 10, Solutions (due date: Mar 30)

Homework 11, Solutions (due date: April 07)

Homework 12 (with the reference for the solutions) (due date: April 15)

Homework 13 (with the reference for the solutions) (due date: April 21)

Prerequisites: Math 532

Alternate course for undergraduates: Math 333

Exams: There will be in class midterm and final. Midterm: March 9th, 8:30 am to 9:45 am and Final: April 18th, 8:30 am to 9:45 am, in the class. 

Gradging: Homework (50%) + Midterm(20%) + Final (30%) 

Topics to be covered:

• 1- Field of Complex Numbers: Fundamental Theorem of Algebra

• 2- Holomorphic Functions: Cauchy-Riemann Equations, Elementary Functions, Mobius Tranfsormations, Harmonic Functions, Conformal Mappings

• 3- Complex Integration: Line Integral, Goursat’s Theorem, Cauchy's Theorem, Cauchy’s Integral Formulas, Morera's Theorem, Schwarz Reflection Principle, Runge's Theorem

• 4- Moremorphic Functions: Zeroes and Poles, Singularities, Residue Formula, Removable Singularity Theorem, Casorati–Weierstrass Theorem, Rouché's Theorem, Open Mapping Theorem, Local Mapping Theorem, Hurwitz's Theorem, Picard Theorems

• Midterm: March 9th, 8:30 am to 9:45 am (Midterm includes Chapters 1,2,3, and 4): Midterm

• 5- Riemann Mapping Theorem and Family of Holomorphic Functions: Compactness Theorems, Montel’s Theorem,

• 6- Analytic Continuation, Gamma Function, Multivalued Extensions, Differential Equations and Multi-Valued Extensions

• 7- A Tase of Riemann Surfaces: Morse Theory, Topology and the Classification of 2-manifolds, Holomorphic Structure, Affine and Projective Varieties.

• 8- Entire functions, Jensen' formula, Infinite Products, and Weierstrass Theorem

• 9- The Zeta function and Prime Number theorem

• Final: April 18th, 8:30 am to 9:45 am 

Other good references:

• Visual Complex Analysis, by Tristan Needham. This is a great book. As the name suggests the book contains many figures and drawings. If you think you need to work more on the basics, this is a good reference.

• Complex Analysis, by Elias M. Stein, Rami Shakarchi. This is another good reference, which contains clear proofs and also good examples. 

• Complex Variables and Applications, 9th edition, by James Brown, Ruel Churchill. It is another good, clear book on the subject. It also contains good exercises and examples.

• Elementary theory of analytic functions of one or several complex variables, by Henri Cartan. A bit old, but very well-written and readable. The book focuses on the basics, and contains very clear proofs of some of the main theorems covered in this course.

• Advanced Complex Analysis, by Curtis T. McMullen. These are notes by McMullen. It contains many interesting examples and questions. It has a geometric taste, with an eye towards the theory of Riemann surfaces and hyperbolic geometry.

• Complex Analysis, 2nd edition, by Eberhard Freitag. This book contains some more advances subjects for further study, including the study of elliptic functions and elliptic modular forms. Furthermore, the 4th chapter of the book on ‘Construction of Analytic Functions’ is a good reference on the subject.

• Classical Topics in Complex Function Theory, by Reinhold Remmert. This book is slightly more advanced than the other references and contains some topics which you can learn after understanding the basics of complex analysis.

• Riemann Surfaces, by Simon Donaldson. We will mention Riemann Surfaces very briefly. This is an interesting book on the subject.

• Lectures on Holomorphic Curves in Symplectic and Contact Geometry by Chris Wendl. A very nice lecture notes. It contains the basics of  complex analysis on C^n, and the applications of holomorphic curve theory to the study of symplectic/contact manifolds. 

• A construction of the Deligne–Mumford orbifold by Joel W. Robbin and Dietmar A. Salamon. A good reference on the Riemann moduli space of Riemann surfaces.

• Pseudo-holomorphic curves and virtual fundamental cycles by John Pardon. This is an informal note on the moduli spaces of holomorphic curves and the geometric structures they carry.