This course consists of lectures and exercise, and contains the basics of the Fourier transform, special functions, partial differential equations, and the Laplace transform.
The aim is for students to be able to use these methods without hesitation when solving physics problems in the future.
At the end of this course, students will be able to apply Fourier transform, special functions, partial differential equations, and Laplace transform to problems in physics.
Fourier transform, gamma function, Legendre functions, Bessel functions, Hermite functions, Laguerre functions, partial differential equations, Green functions, Dirichlet problems, Laplace transform
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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- | - | ✔ | - | ✔ |
In lecture class a (few) report problems may be assigned.
In every exercise class students will be given some problems and solve them.
Some explanations of their solutions will also be given.
Course schedule | Required learning | |
---|---|---|
Class 1 | Review of Fourier expansion and Fourier transform | Understand the Fourier transform as a limit of the Fourier expansion. |
Class 2 | Inverse Fourier transform, Dirac's delta function | Understand the definition of the delta function |
Class 3 | Distribution, application to differential equations | Try solving some differential equations by using Fourier transform |
Class 4 | Gamma function | Understand the definition of the Gamma function |
Class 5 | Stirｌing formula, Beta function | Derive the Stirling formula |
Class 6 | Legendre functions | Derive formulas of Legendre functions from their generating function. |
Class 7 | associated Legendre functions, Spherical harmonics | Understand the relation between associated Legendre functions and spherical harmonics. |
Class 8 | Midterm exam to assess the students’ level of understanding on what has been taught so far and explanation of solutions | Review the course contents so far |
Class 9 | Bessel functions | Derive formulas of Bessel functions from their generating function |
Class 10 | Hankel functions, Neumann functions | Understand the relation between Hankel and Neumann functions and Bessel functions |
Class 11 | modified Bessel functions, spherical Bessel functions | Understand the relation among modified Bessel functions, spherical Bessel functions, and Bessel functions. |
Class 12 | Derive formulas of Hermite and Laguerre functions from their generating functions. | Derive formulas of Hermite and Laguerre functions from their generating functions. |
Class 13 | partial differential equations, Dirichlet problems | Understand the uniqueness of the solution of a Dirichlet problem. |
Class 14 | Green functions | Derive the Green function for the Laplace operator. |
Class 15 | Laplace transform | Understand the relation between Laplace transform and Fourier transform |
Not required
Not required
Based on midterm and final exams, reports, and presentations in exercise classes.
Students are required to have completed Applied Mathematics for Physicists and Scientists I