## Course Title : Mathematics for Electrical and Electronic Engineering １

Code 61020 2nd year 2nd term 2 No Restriction Lecture English Yoshiharu Omura and Shinji Doi

### Course Description

We study properties of eigenfunctions, such as trigonometric functions, Bessel functions, Legendre functions as solutions of linear differential equations, which appear in various subjects of electric and electronic engineering such as electromagnetics, plasma physics, and quantum mechanics. As applications of these eigenfunctions, we also study Fourier series, Fourier transform, and Laplace transform.

The grade will be evaluated based on reports (5points x 13times) and a term examination(100points). If the total points exceed 100 points, the grade is given as 100 points.

### Course Goals

We learn mathematical methods to describe spatial and temporal evolutions of various physical phenomena.

### Course Topics

Theme Class number of times Description
Classification of Partial Differential Equations 2 Partial Differential Equations (PDE) : Laplace, Helmholtz, and diffusion equations; elliptic, hyperbolic, and parabolic types of 2nd order PDE.; derivation of Ordinary Differential Equations (ODE) from PDE by separation of variables
Ordinary Differential Equations 2 Series solutions by Frobenius' method; trigonometric, Bessel, and Legendre functions. Singular points for ODE; Wronskian; linear indepedence of solutions; second solution
Sturn-Liouville Theory 1 Self-ajoint ODE; Hermitian operator; Sturm-Liouville theory
Green's Function Method 1 Green's function method to solve nonhomogeneous equations.
Bessel Functions 2 MATLAB Demonstration (vibrating membrane, EM wave radiation), generating function, Bessel series; application to frequency modulation. Hankel functions; 3D Helmholtz equation in spherical coordinates, spherical Bessel functions
Legendre Functions 1 Legendre functions; generating functions; boundary value problems; associated Legendre polynomials.
Fourier Series 1 Properties of Fourier Series, Gibbs Phenomenon
Fourier Transform 2 Fourier integral, Fourier transforms of Gausian and derivatives, Dirac delta function, Solutions of wave equation and diffusion equation
Laplace Transform 2 Laplace transform, inverse Laplace transform, initial value problems of ODE

### Textbook

Mathematical Methods for Physicists: A Comprehensive Guide, Seventh Edition, Arfken, Weber, and Harris (Kindle version is available.)

### Prerequisite(s)

Calculus, Vector Analysis, Functions of Complex Variable, and English comprehension of the level of VOA Special English