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Complex Variables – Homework for Spring 2018

The assignments are organized according to week they are due. Please report any issues back to Iva.

  • Assignment for Monday 1/22.
  • Assignment for Monday 1/29. You may want to read some of this before attempting the last several problems on the assignment.
  • Assignment for Monday, 2/5.
  • Assignment for Monday, 2/12.
  • Assignment for Monday, 2/19.
  • Assignment for Monday, 3/12.
  • Assignment for Monday, 3/19.
  • Assignment due right after the Spring Break.
  • Assignment due Monday, April 9th.
  • Assignment due Monday, April 16th and the exam study guide.
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    Complex Variables – Spring 2018

    EDUCATIONAL GOALS

  • Goal 1:
    To develop basic literacy involving (functions of) complex variables. This includes:

      – Reaching a degree of fluency with complex number arithmetic and geometry;
      – Developing a solid understanding of elementary functions of complex variable.

    The extent to which this goal is achieved will be determined by the performance on the corresponding midterm exam.

  • Goal 2:
    To become aware of basic ideas of complex analysis. The extent to which this goal is achieved will be determined by the performance on the relevant portion of the second midterm exam.
  • Goal 3:
    To develop a computational ability within the context of calculus of residues. The extent to which this goal is achieved will be determined by the performance of the corresponding midterm exam.
  • Goal 4:
    To gain independence in reading and understanding mathematical material which uses complex variables. The corresponding assessment will be based solely on the take-home final project.
  • Goal 5:
    To situate the practice of complex analysis within its larger mathematical and social context, primarily by understanding the kind of role complex analysis plays in certain currently open problem(s). The corresponding assessment will be based solely on the matching homework assignment.
  •  

    GRADING SCHEME
    For each rubric under Educational Goals (see above) you will receive a letter grade determined by your performance on the corresponding portion(s) of exams, homework assignments etc. In addition, a certain portion of your grade will correspond to the effort you put into participating and staying current in the class; students with a substantial number of class absences and/or late assignments can expect a slightly lowered course grade. The final course grade will be a weighted average of the above:

      Goal 1: 30%
      Goal 2: 5%
      Goal 3: 30%
      Goal 4: 30%
      Goal 5: 5%

    For the description of letter grades and their numerical equivalents please refer to our College Catalog. Please note that a professor has a right to withdraw a student for the reasons of non-attendence.

     

    EXAMS AND SUCH
    There will be two in-class exams, and a partly take-home final exam, each of which will contribute to the course grade through the letter grade for the relevant educational goal. Exams will take place on Monday, February 27th and Monday, April 16th. The role of the final exam will be played by the final project which will be due on Wednesday, May 2nd by 11:30am.

     

    TEXTBOOK
    We will use the 6th edition of Complex analysis for mathematics and engineering by John H. Mathews and Russell W. Howell. Do note that some of the course content will not be directly based on the textbook material. At times, you will be expected to rely on the lecture notes I provide or your own lecture notes.

     

    HOMEWORK
    Most lectures will be followed by a homework assignment, which will be posted online. Homework will be due once a week; most often at class time on Tuesdays. The class meeting prior to the day the homework is due (Monday, in most cases) will be dedicated to answering homework questions or doing extra examples. You are expected to have a draft of your homework completed and with you at that time, and ask homework questions either in class or during subsequent office hours. In particular, I reserve the right to refuse to answer homework questions on the day the homework is due. Note: I have no intention to police due dates or assign numerical values to corrected assignments. Students who are clearly behind (be that mathematically or schedule-wise) will be called in, or in extreme cases: asked to drop the class. Admittance to midterm exams is subject to demonstrated effort to do the homework at the suggested pace. Do me and yourself a favor — take this ……. seriously.

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    Analysis lecture notes

    • Sept 5th:
      We talked about what it takes to axiomatize real numbers, and we started doing the axiomatization. The notes are here.
    • Sept 7th:
      We talked more about axiomatization of real numbers — axioms of order to be specific. Here are the notes. The one and only homework problem is at the very end of the notes.
    • Sept 8th:
      We talked about the Triangle Inequality and just generally strategies for proving inequalities. The homework is at the very end of the notes.
    • Sept 11th:
      Today we introduced the concepts of supremum and infimum. There is one homework problem at the end of the notes.
    • Sept 12th:
      Iva got sick.
    • Sept 14th:
      Homework day.
    • Sept 15th:
      Today we introduced and discussed the necessity of the Completeness Axiom. There are three homework problems at the end of the notes.
    • Sept 18th:
      We work on establishing the existence (and uniqueness) of the n-th root. There is one big mega homework problem at the end of the notes.
    • Sept 19th:
      We continued the discussion from the previous class. There is no new homework.
    • Sept 21st:
      Homework day.
    • Sept 22nd:
      We talked about the Archimedean Property and the Denseness of Rationals. Homework is at the end of the lecture notes.
    • Sept 25th:
      We talked about the exponential function. Homework is at the end of the lecture notes.
    • Sept 26th:
      We continued with the development of the exponential function, and we discussed what one would (in principle) do to develop the logarithmic function.
    • Sept 28th:
      Homework day!!
    • Sept 29th:
      Big day: we introduced the concept of the limit. Homework is at the end of the lecture notes.
    • October 2nd:
      We talked about properties of convergent sequences – arithmetic operations with them, in particular. Homework is at the end of the lecture notes.
    • October 3rd:
      We talked about the properties of limits having to do with the order relation. One homework problem is at the end of the lecture notes, but its due date is …. some other time. Or you can do it for extra credit.
    • October 5th:
      Homework day….
    • October 6th:
      We did the Theorem on Monotone and Bounded Sequences. No due date for the homework is assigned because an exam has been emailed out….
    • October 9th:
      We talked about proving divergence and subsequences. No homework assigned.
    • October 10th:
      We talked about the Bolzano-Weierstrass Theorem. No homework assigned.
    • October 12th:
      In class exam!
    • October 13th:
      Fall break!
    • October 16th:
      We talked about Cauchy sequences. No homework assigned.
    • October 17th:
      Series happened!
    • October 19th:
      Homework day.
    • October 20th:
      More series happened. Homework is at the end of the lecture notes.
    • October 23rd:
      Today we introduced continuity. Homework is at the end of the lecture notes.
    • October 24th:
      We talked more about continuity. Everything worthy of notice is in the lecture notes from yesterday.
    • October 26th:
      Homework day!!
    • October 27th:
      We did the Intermediate Value Theorem.
    • October 30th:
      We did a comparison between miscellaneous concepts of limits, and outlined the need for the concept of a metric space.
    • October 31st:
      Day one of metric spaces!! The homework is here.
    • November 2nd:
      Homework day.
    • November 3rd:
      We talked about the concepts of open and closed sets. The homework is at the end of the lecture notes.
    • November 6th:
      We talked more about open and closed sets; everything said is in the lecture notes posted earlier. The homework is at the end of the lecture notes posted earlier.
    • November 7th:
      We talked more about open and closed sets; everything said is in the lecture notes posted earlier. The homework is at the end of the lecture notes posted earlier.
    • November 9th:
      Homework day!!
    • November 10th:
      We talked about the interior and closure of a (sub)set of a metric space. The notes are coming, but here is the link to the homework assignment.
    • November 13th:
      We talked about sequences in general metric spaces. No new homework was assigned; for notes follow the link below.
    • November 14th:
      We introduced the concept of a complete metric space, and examined how the concept of continuity can be exported into general metric spaces. The homework is at the end of the notes.
    • November 16th:
      Homework day!
    • November 17th:
      We articulated continuity in terms of pre-images of open sets being open.
    • November 20th:
      Completeness of spaces of bounded functions.
    • November 21st:
      Day 2 of completeness of spaces of bounded functions. Here is the new homework assignment!

    Real Analysis: Syllabus for Fall 2017

    Educational Goals

    • Goal 1:
      Get a sense of, and raise the fluency in, foundational aspects of pre-calculus. This includes developing an appreciation for the role the concept of supremum plays underneath all of pre-calculus.
    • Goal 2:
      Gain a deeper appreciation for the concepts of convergence and continuity, through investigations of limits of sequences.
    • Goal 3:
      Develop an abstract mindset which can handle ideas of convergence and continuity from the standpoint of metric spaces.

    The course grade will be determined by the above goals, in a 20-40-40 ratio. The grade for each individual exam will be assigned based on the performance on homework and exams.

    Homework
    Homework will be assigned daily, but collected and graded on a weekly basis. One day a week, namely Thursday, will be set aside for homework discussion. The homework sets will be due on Friday. Note: I have no office hours on Friday.

    Exams
    There will be two midterm exams. They will each have a take-home and an in-class component. The relevant dates are Thursday, October 12th (day before the Fall Break) and Thursday, December 7th. There will be a take-home final exam. The exam will be due at the actual final exam time, Monday, December 18th at 8:30 am.

    Notes
    There is no mandatory textbook for the class. A good book on which you may want to rely is called Real Analysis and is authored by Patrick M. Fitzpatrick. I have some typed up lecture notes from the last time I taught the class. My plan is to update the notes regularly and make them available for you.

    Linear Algebra Educational Goals

    • Goal 1:
      To become fluent in solving linear systems of equations with or without parameters, and to gain an understanding of what the solutions represent geometrically. The corresponding assessment will primarily be based on student performance on the matching in-class exam and homework.
    • Goal 2:
      To become fluent in matrix algebra and determinants, and to gain an understanding of what they represent geometrically. The corresponding assessment will primarily be based on student performance on the matching in-class exam and homework.
    • Goal 3:
      To develop an ability to work with eigenvalues and eigenvectors, both computationally and geometrically. The extent to which this goal is achieved will be determined by student performance on the corresponding exam and homework, as well as the performance on the final exam.
    • Goal 4:
      To internalize the idea of abstract vector spaces. The extent to which this goal is achieved will be determined by student performance on the corresponding homework, as well as the performance on the final exam.
    • Goal 5:
      To become fluent in working with (abstract) inner product spaces. The extent to which this goal is achieved will be determined by student performance on the corresponding homework, as well as the performance on the final exam.
    • Goal 6:
      To be able to do change of basis and coordinates “in your sleep”. The corresponding assessment will primarily be based on student performance on the corresponding homework, as well as the performance on the final exam.

    The course grade will be determined by the above goals, in a 15-15-30-15-15-10 ratio.

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