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.
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
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
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|
|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).|
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)
Students' knowledge of complex functions and special functions will be assessed.
Exercise problems and reports (30 %), and a final examination (70 %).
Students must have successfully completed both Calculus I (LAS.M101 and Calculus II (LAS.M105) or have equivalent knowledge.