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      This forum is available for everyone to post messages to. If you have any questions about the module, please ask them here.

    • Calculus I and II Revision Sheet File
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    • Calculus I and II Revision Sheet - Solutions File
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    • Module Schedule Page

      Each week there will be 3 lecture sessions in total. Session 1 (Tuesday 12-14)  each week is a 2 hour lecture. Session 2 (Friday 10-11) each week is a tutorial, and Session 3 (Friday 9-10)  is a 1 hour lecture. 

      There will be 10 courseworks in total (Tutorial Problem Sets) . These contain important practice questions. Each coursework is roughly associated to the material of the preceding week. It is crucial that you engage with this coursework and that you attempt all problems. If you have any questions regarding a particular point in the coursework or you would like to have a particular problem discussed in Session 2, please let me know ahead of time, preferably via the module's forum. You will submit selected questions from five course works (2, 4, 6, 8, 10), these will contribute to 20% to the final mark of the module (4% each). The coursework will be posted before Sessions 2-3. On the weekend I will announce which questions to submit, and the submission deadline will be on the subsequent Monday (10 days after the coursework was posted). Feedback on the answers  will be provided once the window for solving it has closed. The purpose of the coursework is to help you to engage with the module and to keep up with its content.

      The  handwritten lecture notes will be posted every week.

      Office Hour: 

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  • Week 1


  • Week 2

  • Week 3

  • Module Description

    In the first year module Numbers, Sets and Functions, you met complex numbers and learnt how to perform arithmetic operations on them. What else can be done with complex numbers and can they be used to solve problems entirely about real numbers? In this module we will study how ideas from calculus including continuity, convergence, differentiation and integration extend to the setting of functions defined on the complex numbers. We will see how some definitions transfer directly to the complex setting, but often with significantly different properties and consequences. As an application of this theory, we will see how some complex methods provide a slick way to compute some real integrals and infinite sums.


    Lectures

    Real-time lectures will take place two times a week in hybrid format. A link for lectures appears on the QMplus page, and you can join to see on-screen lectures. You can ask questions via the chat or verbally.

    Tutorials

    There will be one weekly tutorial. In this we will recap key points from the week's lectures, and I will give you more examples to work on, and discuss.

    Assessment

    The final exam  will count for 80% of your module mark.

    The coursework will be structured as follows. Each week I will post two problem sheets:

    Exercise, Week X – formative exercise (not for submission) complementing the material of the lectures. Solutions will not be posted, however, I will be happy to discuss them during office hours.

    Tutorial Problem Set - posted weekly; solutions will be posted a week later. The odd ones are formative, i.e. not for submission. The even ones are summative; please submit these for grading -- each one will count for 4% of your module mark (in total: 20%). Please note that the even exercises may and will rely on the odd ones.



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  • Week 9

  • Week 10

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  • Syllabus

    • Complex numbers, functions, limits and continuity.
    • Complex differentiation, Cauchy-Riemann equations, harmonic functions.
    • Sequences and series, Taylor's and Laurent's series, singularities and residues.
    • Multi-valued functions, branch points
    • Complex integration, Cauchy's theorem and consequences, Cauchy's integral formulae and related theorems.
    • The residue theorem and applications to evaluation of integrals and summation of series.
    • Conformal transformations.


  • Module aims and learning outcomes

    ACADEMIC CONTENT

    This module covers:

    • Complex numbers, functions (single- and multi-valued), limits and continuity.
    • Complex differentiation and integration, including the Cauchy-Riemann equations, harmonic functions, Cauchy’s theorem and Cauchy’s integral formulae.
    • Sequences and series, Taylor’s and Laurent’s series, singularities and residues; the Residue Theorem and applications to evaluation of integrals and summation of series.

    DISCIPLINARY SKILLS

    At the end of this module, students should be able to:

    • Find all complex solutions of a simple polynomial, indicating their position in the Argand diagram.
    • Calculate the derivative of a complex function, explaining where this is well-defined.
    • Explain what is meant by entire, holomorphic and harmonic functions.
    • Derive the Cauchy-Riemann equations for a given function and apply this to determine the set of points in the Argand diagram for which the function is differentiable.
    • Use the theory of Möbius transformations to solve simple problems.
    • Calculate the Taylor and Laurent series of a function about a given point.
    • Find the branch points and define the cut-plane, explaining the choice of a single-valued branch of multi-valued function
    • Define the terms residue and pole, locate them for a given function and calculate their orders.
    • Explain what is meant by a simple pole, a pole of order m and an essential singularity.
    • Define the contour integral of a complex function and evaluate it along a simple contour.
    • State the Residue Theorem and apply it when appropriate to calculate a contour integral.

    ATTRIBUTES

    At the end of this module, students should have developed with respect to the following attributes:

    • Grasp the principles and practices of their field of study.
    • Acquire substantial bodies of new knowledge.
    • Explain and argue clearly and concisely.
    • Acquire and apply knowledge in a rigorous way.
    • Connect information and ideas within their field of study.


  • Assessment information

  • Assessment

    The final exam  will count for 80% of your module mark.

    The coursework will be structured as follows. Each week I will post two problem sheets:

    Exercise, Week X – formative exercise (not for submission) complementing the material of the lectures. Solutions will not be posted, however, I will be happy to discuss them during office hours.

    Tutorial Problem Set - posted weekly; solutions will be posted a week later. The odd ones are formative, i.e. not for submission. The even ones are summative; please submit these for grading -- each one will count for 4% of your module mark (in total: 20%). Please note that the even exercises may and will rely on the odd ones.

     This year exam style will follow that of previous years.

    This year (as in the previous years, see past years exams) you will have to formulate all the theorems and propositions that are used to justify your answer (e.g. Ratio Test, Root Test, the Residue Theorem, the CIF, Cauchy's Theorem, criterion for poles, criterion for removable singularities, Euler's Theorem and more)


  • Teaching team

    • Module organizer: Dr. Mira Shamis

      m.shamis@qmul.ac.uk

  • Exam papers

  • Week 12

  • Q-Review

  • Online Reading List