The subject of this course is complex analysis. Complex analysis is the calculus of complex-valued functions of a complex variable. After reviewing several basic properties of complex numbers, we explain the differentiability of complex functions (holomorphy) and differentiable complex functions (holomorphic functions). Finally, we explain complex line integrals, Green's theorem and Stokes' theorem. In addition, as part of complex analysis, we review contents of [Calculus I / Recitation] and [Calculus II]. This course will be succeeded by [Applied Mathematics for Engineers Ib] in the second quarter.
Complex analysis is an absolutely essential mathematical basis of science and engineering. The aim of this course is to explain the basic theory and practical way to use of complex analysis by an efficient way.
・Students are expected to understand the differentiability of complex functions.
・Students are expected to be familiar with elementary functions as complex functions.
・Students are expected to be able to calculate basic complex line integrals.
・Students are expected to understand the Green's theorem and Stokes' theorem, and be able to apply them to a calculation of an integral.
complex functions, holomorphic functions, complex line integral, Green's theorem, Stokes' theorem
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course mixed with recitation.
Course schedule | Required learning | |
---|---|---|
Class 1 | review on calculus | Details will be announced during each lecture. |
Class 2 | complex numbers and complex plane | Details will be announced during each lecture. |
Class 3 | the differentiability of complex functions, holomorphic functions | Details will be announced during each lecture. |
Class 4 | complex power series | Details will be announced during each lecture. |
Class 5 | complex transcendental functions | Details will be announced during each lecture. |
Class 6 | complex line integral | Details will be announced during each lecture. |
Class 7 | Green's theorem, Stokes' theorem | Details will be announced during each lecture. |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
H. Shiga, Theory of complex functions to learn in 15 weeks (Japanese), Sugakushobo, 2008
None in particular
Based on overall evaluation of the results for quizzes, report and final examination. Details will be announced during a lecture.
This is the prerequisite course to take "Applied Mathematics for Engineers Ib".
Students are expected to have completed [Calculus I / Recitation], [Calculus II] and [Calculus Recitation II].
In particular, students are expected to understand partial differentiation, definite integral and multiple integral clearly.