Foundation of Analysis 2 2024

Time: Thu 1 8:50-10:30

Class Room: 51-07-11

Lecturer: Atsushi Kanazawa

[Description]

This course will cover in a more rigorous manner, and expand the material of the previous course "Complex Analysis". The main gaol is to gain deeper understanding of the theory of complex analytic functions.

[Assignments, Examination & Grade Evaluation]

Final Exam 

[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.

References will be given also during the lectures.

[Q&A session]

Waseda Math Q&A

[Schedule]

4/18 Lecture 1: complex numbers: some function-theoretic and analytic subtleties 

4/25 Lecture 2: power series, radius of convergence, Cauchy-Hadamar's formula

5/2 Lecture 3: complex derivative, holomorphic function, Cauchy-Riemann equations, Wirtinger differential

5/9 Lecture 4: path integral, Green's formula, Cauchy's theorem (rectangular region)

5/16 Lecture 5: Cauchy's theorem, path deformation principle, argument principle, primitive function

5/23 Lecture 6: Cauchy's integral formula, Taylor's theorem, Goursat's theorem, Morera's theorem, identity theorem

5/30 Lecture 7: Cauchy's inequality, Liouville's theorem, fundamental theorem of algebra, maximal principle

6/6 Lecture 8: Schwarz-Pick Lemma, automorphism group of unit disk

6/13 Lecture 9: Laurent series, Laurent expansion

6/20 Lecture 10: isolated singularity, removable singularity, 

6/27 Lecture 11: pole, essential singularity, Weierstrass theorem

7/4 Lecture 12: residue, residue theorem, real integrals revisited

7/11 Lecture 13: point at infinity, Riemann sphere, meromorphic function

7/18 Lecture 14: partial fraction decomposition, linear fractional transformation, etc