In this course, students will practice solving problems related to complex function theory and Fourier series that are widely applicable in science and engineering.
The aim of this course is to aid students’ understanding of Applied Mathematics for Physicists and Scientists I by solving exercise problems.
By the end of this course, students will be able to:
1) Understand the basic concepts and properties of complex functions, such as holomorphicity.
2) Do differentiation and integration of complex functions, as well as integration
of real function by the residue theorem.
3) Solve the boundary value problem of 2-D Laplace’s equation by using conformal mapping technique.
complex function, holomorphy, Cauchy’s internal theorem, residue theorem, conformal map, analytic continuation, Fourier series
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
This course follows the progress of Applied Mathematics for Physicists and Scientists I. Exercise problems will be assigned every class meeting and will be due the next class. Problems are to be solved at the blackboard in the classroom, or submitted as written assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | Complex numbers | To understand the concept of complex numbers. |
Class 2 | Holomorphic functions | To understand the properties of holomorphic functions. |
Class 3 | Elementary functions | To learn about elementary functions defined for complex numbers. |
Class 4 | Complex integration 1 | To learn how to perform complex integration. |
Class 5 | Complex integration 2 | To learn how to perform complex integration. |
Class 6 | Power series | To understand power series expansion of complex functions. |
Class 7 | Residue theorem | To understand Cauchy’s theorem and residue theorem. |
Class 8 | Complex integration 1 | To learn how to perform complex integration. |
Class 9 | Complex integration 2 | To learn how to perform complex integration. |
Class 10 | Conformal mapping | To understand the concept of conformal map and learn conformal map of elementary functions. |
Class 11 | Application of conformal mapping | To learn how conformal mapping can be used in physics. |
Class 12 | Analytic continuation | To understand the concept of Analytic continuation. |
Class 13 | Riemann surface | To understand the concept of Riemann surface and learn the structures of Riemann surfaces for several types of multi-valued functions. |
Class 14 | Fourier series and Fourier transformation | To understand the properties of Fourier series and learn how to perform Fourier transformation. |
See the page of Applied Mathematics for Physicists and Scientists I.
See the page of Applied Mathematics for Physicists and Scientists I.
The course evaluation is based on in-class assignment, take-home written assignment and a term-end examination.
Enrollment in Applied Mathematics for Physicists and Scientists I is desirable.