2025 (Current Year) Faculty Courses School of Science Undergraduate major in Earth and Planetary Sciences
Mathematics for Physics B (EPS course)
- Academic unit or major
- Undergraduate major in Earth and Planetary Sciences
- Instructor(s)
- Masaki Matsushima
- Class Format
- Lecture/Exercise
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- EPS.B210
- Number of credits
- 110
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This course teaches complex functions and special functions in mathematics for physics.
Students learn complex functions and special functions required to understand physical phenomena and physical processes related to Earth and Planetary Sciences.
Course description and aims
By completing this course, students will be able to
(1) Understand the theorem of residue correctly, and carry out complex integrals using the theorem.
(2) Understand special functions, and solve physical problems related to Earth and Planetary Sciences by using them.
Keywords
Mathematics for physics, Complex function, Regularity of complex function, Cauchy's integral formula, Theorem of residue, Special function, Bessel function, Legendre function
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
At the beginning of each class, solutions to exercise problems assigned during the previous class are reviewed. In class, students are given exercise problems related to the lecture given that day to solve.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Complex number, Complex plane, Complex function, Riemann surface | Express complex numbers on a complex plane. Express complex numbers in terms of the polar form. Explain a many-valued function. Explain the Riemann surface. |
| Class 2 | Differentiation and regularity of complex function | Explain differentiability of a complex function. Explain the Cauchy-Riemann's equations. |
| Class 3 | Cauchy's integral theorem | Explain the Cauchy's integral theorem. |
| Class 4 | Cauchy's integral formula, integral representation for derivatives | Calculate a complex integral using Cauchy's integral formula. Explain the Goursa's theorem related to integral representation for derivatives. |
| Class 5 | Power series expansion and its application | Explain the Taylor expansion and the Laurent expansion. |
| Class 6 | Theorem of residue (1) | Calculate various definite integrals using the theorem of residue. |
| Class 7 | Theorem of residue (2) | Calculate various definite integrals using the theorem of residue. |
| Class 8 | Gamma function and Beta function | Explain analytical features of the Gamma function and Beta function. |
| Class 9 | Basic of special function | Explain solutions of the Laplace equation in terms of variable separation. |
| Class 10 | Bessel function (1) | Explain various features of the Bessel function. |
| Class 11 | Bessel function (2) | Explain physical phenomena related to Earth and Planetary Sciences using Bessel functions. |
| Class 12 | Legendre function (1) | Explain various features of the Legendre function. |
| Class 13 | Legendre function (2) | Explain physical phenomena related to Earth and Planetary Sciences using Legendre functions. |
| Class 14 | Spherical harmonic functions | Explain various features of spherical harmonic functions. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend the appropriate amount of time preparing for class and the other appropriate amount of time reviewing class content afterwards (including assignments) for each class.
They should do so by referring to reference books and other course material.
Textbook(s)
None required.
Reference books, course materials, etc.
Fukuyama, Hidetoshi and Ogata, Masao, "Mathematics for Physics, I" Asakura Shoten, ISBN 978-4-254-13703-3 (Japanese).
Shiga, Hironori, "Complex Function Theory" Sugaku-shobou, ISBN 978-4-903342-03-0. (Japanese)
Evaluation methods and criteria
Students' knowledge of complex functions and special functions will be assessed.
Exercise problems and reports (30 %), and a final examination (70 %).
Related courses
- EPS.B201 ： Mathematics for Physics A (EPS course)
Prerequisites
Students must have successfully completed both Calculus I (LAS.M101 and Calculus II (LAS.M105) or have equivalent knowledge.
Other
We may use GNU Octave known as a scientific programming language to deeply understand special functions.