2016 Mathematics for Physics B (EPS course)

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Academic unit or major
Undergraduate major in Earth and Planetary Sciences
Nakajima Junichi  Matsushima Masaki 
Class Format
Lecture / Exercise     
Media-enhanced courses
Day/Period(Room No.)
Mon7-8(I311)  Thr7-8(I311)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
Access Index

Course description and aims

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.

Student learning outcomes

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.


Mathematics for physics, Complex function, Regularity of complex function, Cauchy's integral formula, Theorem of residue, Special function, Bessel function, Legendre function

Competencies that will be developed

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 that were assigned during the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Complex number, Complex plane Express complex numbers on a complex plane. Express complex numbers in terms of the polar form.
Class 2 Complex function, Riemann surface Explain a many-valued function. Explain the Riemann surface.
Class 3 Differentiation and regularity of complex function Explain differentiability of a complex function. Explan the Cauchy-Riemann's equations.
Class 4 Complex power series Explain a complex power series and its radius of convergence.
Class 5 Cauchy's integral theorem, Cauchy's integral formula Explain the Cauchy's integral theorem. Calculate a complex integral using Cauchy's integral formula.
Class 6 Integral representation for derivatives Explain the Goursa's theorem related to integral representation for derivatives.
Class 7 Power series expansion and its application Explain the Taylor expansion and the Laurent expansion.
Class 8 Theorem of residue Calculate various definite integrals using the theorem of residue.
Class 9 Basic of special function Explain solutions of the Laplace equation in terms of variable separation.
Class 10 Gamma function and Beta function Explain analytical features of the Gamma function and Beta function.
Class 11 Bessel function Explain various features of the Bessel function.
Class 12 Legendre function (1) Explain various features of the Legendre function.
Class 13 Legendre function (2) Explain various features of spherical harmonic functions.
Class 14 Application of special functions (1) Explain physical phenomena related to Earth and Planetary Sciences using special functions (1).
Class 15 Application of special functions (2) Explain physical phenomena related to Earth and Planetary Sciences using special functions (2).


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)

Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed both Calculus I (LAS.M101 and Calculus II (LAS.M105) or have equivalent knowledge.

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