Section outline

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    General

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    Course Lecturer:  Dr. Michael Oyiengo Obiero  Email: obiero@maseno.ac.ke
    Technical Support:  Juma Zevick Otieno   

    Email: jumazevick@gmail.com

    WhatsApp:+254701677178
    Duration: 60Hours
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      Course Information

      Introduction

      This is a third year second semester course and is offered to those who have done MMA 300 Real Analysis I. The course begins with a brief introduction to differential calculus of complex valued functions and is extended to integral calculus of the same functions via Cauchy integral theorem and formula together with Residue theorem. Finally the course entails Taylor and Laurent series for complex valued functions. 

      The Course will be offered online and there will also be a mid-semester face-to-face meeting between the course instructor and the other course participants. The participants are required to be active most of the times and to engage the instructor whenever you do not understand a concept.

      The main prerequisites for the course are MMA 100 Basic Mathematics, MMA 103: Linear Algebra, Introduction to Analysis and Real Analysis I, and Calculus I and II.

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      Course Requirement

      This course will require at least 60 hours for effective coverage of the course content. That translate to at least 6 hours of your time per week. Besides, each topic has lecture notes together with the exercises for the participants which you are required to solve and submit every week. Late submission is not allowed. In addition to that you will be required to have the following:

      • A laptop/desktop or smartphone together with internet connection.
      • A scientific calculator and,
      • A pen and a writing pad.

      The course will be evaluated through continuous assessment tests and end semester examination.

      • Mastery quizzes 15%
      • Test quizzes 15%
      • End of Semester exam 70%.

      Pass mark for the course is 40% of the total score.

      Once opened, Mastery quizzes will remain open for the duration of the course and can be attempted as many times as you wish. Test quizzes on the other hand will have limited availability.

      There will be no extensions to deadlines. Plan your time well to meet  the set deadlines.

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      Course Objectives

      By the end of this course you should be able to:

      1. Define limits of complex valued functions and compute the limits of various functions at various points.
      2. Explain the meaning of continuous functions at various points in the complex plain.
      3. Define the derivative of a complex valued function and use these derivatives to determine whether a function is Harmonic or not.
      4. State and prove Cauchy's integral theorem and apply it to compute the integral of various function within certain regions (contours).
      5. Compute the Taylor and Laurent series of complex valued functions within some region.
      6. Apply Residue theorem to integrate functions that can be expressed in Laurent series.

    • Course Resources

      The main reference for this course is the book "A first course in Complex Analysis with Applications" by Dennis Zill and Patrick Shanahan.

      Any other relevant resources are encouraged. There a re lots of online videos on Complex analysis. Some of these will be provided as links in the course. There are also lots of online notes on Complex Analysis. Feel free to explore these resources.

    • COMPLEX ANALYSIS I COURSE TEXT File
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    Week 1- Complex Numbers and the Complex Plane

    You might have come across complex numbers in elementary courses learned before. You might also be aware of some properties of complex numbers. In this topic, we introduce complex numbers and review some basic arithmetic calculations involving complex numbers.

    Read sections 1.1 and 1.2 of our reference text. 

    • Objectives

      By the end of this topic you should be able to:

      1. Define and classify complex numbers.
      2. Add, subtract, divide and multiply complex numbers.
      3. Multiply and divide complex numbers.
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      Learner Activities

      • Download and Refer to the Course BOOK for a detailed understanding of the topic
      • Watch the video tutorial for a better understanding of the course content
      • Engage with your colleagues in the discussion forum.
      • Attempt the Online STACK Quiz
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      You might also want to watch  

       

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      Learning Resources

      Below is a video on multiplying complex numbers.

      .

    • MASTERY QUIZ 1
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    • TIMED QUIZZ 1
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    • Discussion Forum 1
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    Week 2- Polar Forms , Powers and Roots of Complex Numbers

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      Introduction

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      Objectives

      By the end of this topic you should be able to:

      1. Express a complex number in polar form.
      2. Compute Powers of a Complex Number.
      3. Calculate roots of a complex number.

      Learning Activities

      • Read on sections 1.2, 1.3 and 1.4 of our course reference book.
      • Watch the video tutorials for a better understanding of the course content
      • Engage with your colleagues in the discussion forum.
      • Attempt the weekly quizzes
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      Tutorial Videos

      Writing a complex number in polar form

      Powers and roots of complex numbers

    • MASTERY QUIZ 2
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    • TIMED QUIZ 2
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    • Discussion Forum 2
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    WEEK 3- Complex functions and limits

    In this topic, we review ideas of functions and develop the concepts of complex functions. We compare with the concepts of functions of real variables. The topic also discusses the concept of limits of a complex function and highlights the main differences with limits of functions of real variables.

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      Objectives

      By the end of this topic, you should be able to:

      1. Define a complex function and deduce the real and imaginary parts of a complex function.
      2. Evaluate the values of complex functions at a given point.
      3. Determine whether a complex function has a limit at a point or not.
      4. Evaluate limits of complex functions at a given point.

      Learning activities

      1. Read on sections 2.1 through to 2.6 of the course reference book.
      2. Read additional lecture notes provided below.
      3. Watch tutorial videos on complex functions to enhance your understanding.
      4. Attempt the weekly quizzes.
      5. Engage your colleagues in group discussions on the forum.

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      Introduction to complex functions

      Limits of a complex function

    • Discussion Forum 3
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    • MASTERY QUIZ 3
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    • TIMED QUIZ 3
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    WEEK 4 - Limits, continuity and analytic functions

    In this topic, we use the concept of the limit of a complex function to define continuity of a complex function. We also introduce the concept of complex differentiation. Cauchy Riemann equations will be introduced and used to prove analyticity of complex functions.

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      Objectives

      By the end of this topic you should be able to:

      1. Determine whether a complex function is continuous in a region or not.
      2. State the Cauchy Riemann equations.
      3. Determine whether a certain complex valued function is analytic or not.
      4. Compute the derivatives of complex functions.
      5. Determine whether a function is harmonic or not.
      6. Calculate harmonic conjugates of harmonic functions.

      Learning activities

      1. Read section 2.6 and chapter 3 of the course reference book.
      2. Read additional lecture notes provided below.
      3. Watch tutorial videos to enhance your understanding of concepts covered in this topic.
      4. Attempt weekly quizzes for this topic within the time limits.
      5. Engage in discussions with colleagues on the forum.
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      Learning Activities

      1. Download Topic 3 Lecture Notes and also the extra material and read.
      2. Solve exercises in Topic 3 Lecture Notes and submit your results.
      3. Read pages 69-73 of Course Book 1 and 32-36 of Course Book 2 and make short notes.
      4. Browse web page en.wikipedia.org/wiki/Harmonic_function.
      5. Engage your colleagues in discussion.
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      Learning Resources

      Analytic functions

    • Discussion forum 4
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    • MASTERY QUIZ 4
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    • TIMED QUIZ 4
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    WEEK 5 - Analytic functions and Harmonic functions

    This week we continue with the study of analytic functions. Our focus will be on the theorem

    Theorem: A complex valued function f(z)=u(x,y) + i v(x,y) is analytic if and only if it satisfies the Cauchy-Riemann equations.

    We also study Harmonic functions and how they relate to analytic functions. A function u(x,y) of real variables is said to be harmonic if it satisfies the Laplace's equation.

    Recall our previous encounter with Laplace's equations in MMA 202: Vector Analysis?

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      Learning Objectives

      By the end of this topic you should be able to:

      1. Determine whether a function is analytic or not.
      2. Determine the domain of analyticity of an analytic function.
      3. State and use the Cauchy Riemann equations to determine whether a function is analytic.
      4. State the Laplace's equation and use it to determine whether a function is harmonic.
      5. Evaluate the harmonic conjugate of a harmonic function.

      Learning activities

      1. Read Chapter 3 sections 3.1 and 3.2 on analytic functions and attempt exercises.
      2. Read section 3.3 of the course reference book on harmonic functions and attempt exercises.
      3. Read additional uploaded notes below.
      4. Attempt the Mastery and Timed quizzes.

      Below is a video on how to deal with the complex exponential function.

    • Harmonic functions File
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    • Laplace equation and harmonic functions File
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    • Discussion forum 5
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    • MASTERY QUIZ 5
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    • TIMED QUIZ 5
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    WEEK 6 - Integration in the complex plane

    This week we start on the study of integration in the complex plane. There are some similarities with integration involving real variables, but as you will find out, the integrals here cover a much wider class of functions. We will also introduce (and use) techniques and formulas for integrating in the complex plane.

    Students are encouraged to read section 5.1 of the course lecture notes to review on Real integrals.

    The main theme in integration in the complex plane is contour integrals (integrating over a curve). Techniques for evaluating the integral will depend on the nature of the curve, and the function being integrated with respect to the said curve.

    • Learning Objectives

      By the end of this topic you should be able to:

      1. Evaluate contour integrals by reparameterization of curves.
      2. Use properties of contour integrals in calculating integrals.
      3. Evaluate integrals over piecewise smooth curves.
      4. Determine bounds of contour integrals.
      5. Apply the Cauchy Goursat theorem in calculating contour integrals.
      Learning activities
      1. Read section 5.1 of the course lecture notes to review on Real integrals
      2. Read section 5.2 of the lecture notes on contour integrals
      3. Read on section 5.3 of the lecture notes on Cauchy Goursat theorem and its application in solving contour integrals.
      4. Download and read additional lecture notes provided below.
      5. Attempt exercises in the course reference book and thee weekly quizzes. 

      Watch the videos below for an in-depth explanation of contour integrals.

    • Learning Resources

    • Topic 6 Lecture Notes File
    • Extra material File
    • Discussion forum
    • MASTERY QUIZ 6
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    • TIMED QUIZ 6
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    WEEK 7 - Contour Integration and Series

    In this topic we continue with the study of contour integrals. In particular we apply the Cauchy Integral Formula and The Cauchy Integral Formula for derivatives to solve complex integrals.

    We also have a brief study of Taylor series and Laurent Series of complex functions. 

    In computing Laurent series, in most cases the Laurent theorem is not applied but we do use the other easily available techniques to determine the coefficients of the series. These techniques are the binomial expansion and the Taylor series expansion. This is because of the simplicity of these techniques in comparison to Laurent theorem.

    Learning Objectives

    By the end of this topic you should be able to:

    • Evaluate complex integrals using the Cauchy Integral Formula.
    • Compute the Taylor series of various functions.
    • Determine the radius of convergence of various  series.
    • Compute Laurent series of a function valid for a specific region.
    Learning Activities
    • Read section 5.5 of our course reference book and attempt the exercises.
    • Read sections 6.2 and 6.3 of our course reference book on Taylor series and Laurent series.
    • Download and read the attached lecture notes below.
    • Watch the video on Taylor and Laurent series for additional information.

    • Learning Resources

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    • MASTERY QUIZ 7
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    • TIMED QUIZ 7
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    • Computation of Laurent series Forum
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    WEEK 8 - Singularities, Residues and the Residue theorem

    Highlighted

    We start by the study of singularities of complex integrals. We classify the three types of singularities from the Laurent series expansion of the complex function.

    For a case when the singularities are poles, we use the Laurent series expansion of the complex function to determine the residue of the function at that particular point. The residues are then used to evaluate the integral of the complex function. This is the study of the residue theorem.

    • Learning Objectives

      By the end of this topic you should be able to:

      1. Compute the residue of analytic functions with finite number of poles of various orders.
      2. State the Residue theorem.
      3. Use the Residue theorem to evaluate integrals.

    • Learning Activities

      1. Read sections 6.4 and 6.5 of the course reference book.
      2. Attempt exercises in sections 6.4 and 6.5 of the course reference book.
      3. Watch additional videos on the Residue theorem..
      4. Attempt the weekly homework.

    • Topic Summary

      When analytic function has more than one pole in a certain closed curve then the best way to find its integral over that curve is to use Residue theorem since it will be easy to compute the function's residue at each and every pole.

    • Learning Resources

    • Topic 8 Lecture Notes File
    • Extra Notes File
    • Residue theorem Forum
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    CHOOSING YOUR REVISION GROUP

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    EXAM REVISION SECTION- GROUPS

    • EXAM REVISION QUESTIONS - GROUP ONE (A) Quiz
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    • EXAM REVISION QUESTIONS - GROUP ONE (B) Quiz
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    END OF SEMESTER QUESTIONNAIRE