Complex Analysis

Vrije Universiteit Amsterdam

Course Description

  • Course Name

    Complex Analysis

  • Host University

    Vrije Universiteit Amsterdam

  • Location

    Amsterdam, The Netherlands

  • Area of Study

    Mathematics

  • Language Level

    Taught In English

  • Course Level Recommendations

    Upper

    ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.

    Hours & Credits

  • Recommended U.S. Semester Credits
    3
  • Recommended U.S. Quarter Units
    4
  • Overview

    COURSE OBJECTIVE
    - The student can decide whether a complex function is analytic (=differentiable in the complex sense) and knows the connection with the
    Cauchy-Riemann equations.
    - She can do computations with elementary functions such as exp/log/sin/cos over the complex numberts.
    - She can integrate analytic functions along a path on the complex plane, using the theorem of Cauchy-Goursat and its corollaries.
    - She can compute Laurent series and determine the type of singularities of analytic functions.
    - She can compute integrals of complex functions using the residue theorem and knows how to use this to compute integrals of real
    functions.

    COURSE CONTENT
    In complex analysis one generalizes the standard concepts of real analysis such as differentiation and integration from the real line to the complex plane. Although these generalizations arise very naturally and all standard examples of functions are also differentiable in the complex sense, the latter property surprisingly turns out to be much stronger. As a consequence, complex differentiable functions immediately obey very special properties which we are going to explore in this course. In particular, they lead to completely new and efficient methods for computing integrals of real functions.

    During the lectures the following topics will be treated:
    - complex differentiation and the Cauchy-Riemann equations
    - complex integration and the theorem of Cauchy-Goursat
    - elementary properties of complex differentiable functions
    - singularities, Laurent series and the residue theorem
    - application to integrals of real functions

    TEACHING METHODS
    Lecture (2 hours) and tutorial class (2 hours)

    TYPE OF ASSESSMENT
    Two written exams (40%+40%) and two hand-in homeworks (10%+10%). The retake exam counts for 100% of the final grade.

    RECOMMENDED BACKGROUND KNOWLEDGE
    Calculus, Analysis, Linear algebra

Course Disclaimer

Courses and course hours of instruction are subject to change.

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