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Math 407
Fall 2017 Journal

Monday, December 11
I posted solutions to the final examination.
Friday, December 8
The final examination was given.
Tuesday, December 5 (redefined as Thursday)
After some announcements, we worked on four review exercises, shown on the last page of the slides from class. I have posted solutions.
Thursday, November 30
We discussed some methods for computing residues and worked an example of applying the residue theorem to evaluate a real calculus integral by converting the problem into a complex line integral over a closed path. Then we took a quiz in groups, which turned into a take-home quiz for the groups that did not finish during class.
If you missed class today, then you may work the quiz as an individual assignment and turn it in on Tuesday. The quiz is the last page of the slides from class.
Tuesday, November 28
We revisited Laurent series, discussed the concept of the residue of an analytic function, and worked some examples. The assignment to hand in next time, as shown on the last page of the slides from class, is parts (a), (b), and (c) of Exercise 1 in Section VII.1.
Tuesday, November 21
We revisited the radius of convergence of power series and then applied the formula for the sum of a geometric series to deduce Taylor’s formula from Cauchy’s integral formula for derivatives. And we looked at an example of a Laurent series. The slides from class are available.
Thursday, November 16
We continued the discussion of determining the radius of convergence of a power series, and we worked some more examples as a group quiz. The slides from class are available. Here is the assignment for next time (some review exercises):
  1. When is the final exam?
  2. Determine all values of the integer \(n\) for which \( i^n=1\).
  3. Determine all values of the complex number \(z\) for which \(e^z =1\).
Tuesday, November 14
We discussed convergence of infinite series of complex numbers and the notion of radius of convergence of a power series. As shown on the last page of the slides from class, the assignment to hand in next time is parts (b), (c), and (f) of Exercise 1 in Section V.3.
Thursday, November 9
We discussed Cauchy’s integral formula and looked at some examples. As shown on the last page of the slides from class, the assignment to hand in next time is parts (a), (b), (c), and (e) of Exercise 1 in Section IV.4.
Tuesday, November 7
We discussed path independence of integrals. As shown on the last page of the slides from class, the assignment to hand in next time is Exercise 2 in Section III.2 and Exercise 4 in Section IV.1.
Thursday, November 2
We discussed the notions of closed differentials and exact differentials and the interpretation of these concepts in the settings of analytic functions and harmonic functions. We started working in class on the assignment (due next time), which is shown on the last page of the slides from class.
Tuesday, October 31
We discussed line integrals, Green’s theorem, and Cauchy’s integral theorem. As shown on the last page of the slides from class, the assignment due next time is all three parts of Exercise 1 in Section III.1.
Thursday, October 26
The second exam was given, and solutions are available.
Tuesday, October 24
We discussed the notion of the point at infinity in connection with linear fractional transformations and worked a quiz in groups. The slides from class are available, along with solutions to the quiz.
There is no assignment to hand in next time, since the second exam takes place in class on Thursday, October 26.
Thursday, October 19
We discussed the concepts of conformal mappings and Möbius transformations. I have posted what the slides might have been if the computer had been working.
In view of the exam that takes place in class on Thursday, October 26, there is nothing to hand in on Tuesday, October 24. The assignment is to begin reviewing and preparing for the exam.
Tuesday, October 17
We followed up on the topics of analytic functions, harmonic functions, harmonic conjugates, and local invertibility of analytic functions. As shown on the last page of the slides from class, the assignment to hand in next time is Exercise 5 in Section II.5 about the Laplacian in polar coordinates.
Thursday, October 12
We discussed the implications of the Cauchy–Riemann equations and the Jacobian matrix for local invertibility of analytic functions. And we took a quiz in groups.
The assignment due next time, as shown on the last page of the slides from class, is the following.
  • Section II.3, Exercise 1(b)
  • Section II.4, Exercise 2
  • Section II.5, Exercise 1(f)
Tuesday, October 10
We discussed the notions of analytic functions, harmonic functions, and harmonic conjugates. In groups, we solved an exercise on computing a harmonic conjugate function.
As shown on the last page of the slides from class, the assignment to hand in next time is Exercise 3 of Section II.3 and Exercise 2 of Section II.5.
Thursday, October 5
We continued the discussion of complex derivatives and the Cauchy–Riemann equations, and we saw that the real part and the imaginary part of a complex differentiable function have orthogonal level curves.
As shown on the last page of the slides from class, the assignment to hand in next time is Exercises 1(c) and 3 from Section II.2 and Exercise 2 from Section II.3.
Tuesday, October 3
I returned the graded exams. We discussed the notion of the complex derivative and the Cauchy–Riemann equations.
The assignment to turn in next time is on the last page of the slides from class.
Thursday, September 28
The first exam was given, and solutions are available.
Tuesday, September 26
We reviewed for the exam to be given next class, and we worked a quiz in groups. I have posted solutions to the quiz.
Thursday, September 21
We discussed the representation of the trigonometric functions in terms of the exponential function, the corresponding formulas for the hyperbolic functions, and the strategy for computing a formula for the inverse cosine function. We worked together on some exercises about the trigonometric and hyperbolic functions, and I have added solutions to the slides from class.
Since the first exam takes place on Thursday, September 28, the assignment (not to hand in) is to make yourself flash cards for all the main concepts and formulas from Chapter I.
Tuesday, September 19
We discussed the meaning of \(z^w\) when \(z\) and \(w\) are complex numbers, and we worked on Exercise 1 in Section I.7. The slides from class are available.
The assignment to turn in next time is parts (a) and (b) of Exercise 1 in Section I.7. Notice that the answer is in the back of the book, so you need to show work.
Thursday, September 14
We discussed the complex logarithm function and the notion of different branches. We worked together on Exercise 1 in Section I.6. Then we took a quiz in groups (see the last page of the slides from class). Solutions are available.
Here is the assignment to turn in next class:
  • Section I.6: Exercise 2(a),(b)
  • Section I.4: Exercise 1(b),(c)
  • Section I.1: Exercise 1(i)
Tuesday, September 12
We discussed issues about defining inverse functions of complex functions, especially the square-root function. The colored picture of \(z^2\) that we looked at was built using David Bau’s complex function viewer.
The assignment to hand in next time is from Section I.5, Exercise 2, parts (a) and (b). The slides from class are available.
Thursday, September 7
We discussed some structural properties of the complex numbers and the polar form of complex numbers. Then we worked together on Exercise 1 in Section I.5.
The assignment to hand in next class is on the final page of the slides from class. Also available are solutions to the take-home quiz that you turned in today.
Tuesday, September 5
We discussed notation and terminology for complex numbers; the definition of the exponential, sine, and cosine functions for complex numbers; and Euler’s formula. We also looked at Exercises 3, 5, and 7 from Section I.1.
The assignment to hand in at the beginning of the next class (Thursday, September 7) is the Quiz at the end of the slides from class.
Thursday, August 31
On this first day of class, we discussed the complex numbers and worked examples of computing reciprocals and square roots. The notes from class are available.
The assignment is to read section I.1 in the textbook and to find an exercise at the end of that section that you don’t know how to solve. There is no homework to hand in next class, but you should be prepared for the possibility of a quiz.
Tuesday, August 22
This site went live today. Once the Fall 2017 semester begins, there will be regular updates about assignments and the highlights of each class meeting.