Foundation of Analysis 2 2026

Time: Fri 1 8:50-10:30

Class Room: 51-711 51-17-08

Lecturer: Atsushi Kanazawa

[Description]

Complex analysis is a particularly beautiful and powerful branch of mathematical analysis. It was shaped by the pioneering work of mathematicians such as Euler, Gauss, and Riemann, and continues to play a vital role in both pure and applied mathematics today. This course provides an introduction to the basic theory of complex functions, focusing on essential topics such as analytic functions, complex integration, and power series. While we will follow a standard approach, we will also highlight connections to applications in physics and engineering. The course avoids unnecessary abstraction and aims to develop both conceptual understanding and computational skills.

[Assignments, Examination & Grade Evaluation]

reports 80% + attendance 20%

[Materials & Reading List]

There is no fixed textbook, but participants are encouraged to consult online textbooks on "Complex Analysis" or "Function Theory" which are available for download from Waseda Library. 

Lecture note will be shared on Moodle. 

[Q&A session]

Waseda Math Q&A

[Schedule]

4/12 Lecture 1: complex plane, topology

4/19 Lecture 2: complex functions: exponential function, logarithmic function, power function

4/26 Lecture 3: complex derivative, Cauchy-Riemann equation

5/10 Lecture 4: holomorphic functions: examples and basic properties 

5/17 Lecture 5: series, absolute convergence, convergence test, power series

5/24 Lecture 6: curve, line integral

5/31 Lecture 7: primitive function, Cauchy's theorem

6/7 Lecture 8: uniform convergence, Cauchy's Integral formula

6/14 Lecture 9: Taylor expansion, identity theorem

6/21 Lecture 10: Liouville's theorem, maximal principle, Schwarz's lemma

6/28 Lecture 11: Laurent series, isolated singularity

7/5 Lecture 12: removable singularity, pole, essential singularity

7/12 Lecture 13: residue, residue theorem, real integrals revisited

7/19 Lecture 14: related advanced topics