This course focuses on the complex function theory and Fourier series widely applicable to the field of science and engineering.
This course has two aims. The first is to understand the derivative and integral of the complex functions. The other is to understand the basic of the Fourier analysis.
By the end of this course, students will be able to:
1) explain the basic concept of the complex function theory.
2) understand the derivative and integral of the complex functions and calculate the integral of the real function by means of the residue theorem.
3) explain the conformal map for the holomorphic function and solve two-dimensional Laplace equations.
4) explain the concept of the analytic continuation.
5) explain the concept of the Fourier series for the periodic functions and obtain the series coefficients.
complex function, holomorphy, Cauchy's integral theorem, residue theorem, conformal map, analytic continuation, Fourier series
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
The complex function theory is learned in the class in terms of the textbook. To prepare for class, students should read the course schedule section and check what topics will be covered. Required learning should be completed outside of the classroom for preparation and review purposes.
|Course schedule||Required learning|
|Class 1||complex variables||Compute the operations of complex variables.|
|Class 2||holomorphic function||Understand the holomorphic functions|
|Class 3||elementary functions||Understand the elementary functions.|
|Class 4||complex integral 1||Understand the contour integral in the Gauss plane|
|Class 5||complex integral 2||Understand the Cauchy's theorem|
|Class 6||power series||Compute the series coefficients.|
|Class 7||residue theorem||Understand the residue theorem|
|Class 8||application of complex integral 1||Compute the integral for the real function by means of the complex integral.|
|Class 9||application of complex integral 2||Compute the integral for the real function by means of the complex integral.|
|Class 10||Conformal map||Understand the conformal map|
|Class 11||application of conformal map||Solve the two-dimensional Laplace equations|
|Class 12||analytic continuation||Understand the identity theorem and analytic continuation.|
|Class 13||Riemann surface||Understand the Riemann surface|
|Class 14||Fourier series||Compute the Fourier coefficients|
|Class 15||Fourier transformation||Perform the Fourier transformation|
R. V. Churchill and J. W. Brown, Complex valuables and applications, Sugaku Shobo (Japanese)
Shinichi Oishi Fourier Analysis Iwanami Shoten
Students’ course scores are based on final exams.
No prerequisites are necessary, but enrolment in the related exercises is desirable.