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Math 407
Fall 2016 Journal

December 12
I posted final grades at eCampus.
December 9
The final examination was given, and solutions are available.
December 6
At this last class meeting, we looked at problem 7 from the Fall 2011 final exam, we tried to compute the hard integral from the first day of class—a complete solution is available—and we reviewed for the final examination, which takes place on Friday, December 9, from 12:30 to 2:30.
December 1
We worked in groups on a quiz, for which solutions are available.
The assignment for next time (not to hand in) is to make a list of the main topics covered in the course. The final class meeting is on Tuesday, December 6, and the final examination is on Friday, December 9, from 12:30 to 2:30.
November 29
We continued the discussion of residues, including methods for finding residues at simple poles and at higher-order poles. Then we applied the residue theorem to prove that \[\int_0^\infty \frac{1}{1+x^4}\,dx = \frac{\pi}{2\sqrt{2}}. \]
The assignment to hand in next time is Exercise 5 on page 202 in section VII.2.
November 22
We discussed residues and the residue theorem, and we worked some examples of computing residues and integrals.
The assignment for over the Thanksgiving break is to be safe.
November 17
We continued the discussion of series expansions of functions, including Laurent series, and saw the connection between principal parts and the partial-fractions decomposition of a rational function.
Here is the assignment to hand in next time:
  • Exercise 2(a) on page 176 in section VI.2. (Notice that the singularity where \(z=1\) is removable!)
  • Exercise 1(b) on page 170 in section VI.1. (There is a series in positive powers of \(z\) that converges when \(|z| \lt 1\) and a series in negative powers of \(z\) that converges when \(|z| \gt 1\).)
  • Exercise 7 on page 10 in section I.2. (This problem from Chapter I is relevant to the upcoming topic of computing integrals via the residue theorem and also is a reminder that starting to review for the final exam is a good idea.)
November 15
We discussed the classification of isolated singularities (poles, removable singularities, essential singularities).
The assignment to hand in next time is to find the principal part at the singular point where \(z=1\) for the functions in Exercise 1 parts (a) and (b) on page 176 in section VI.2. (There are poles also when \(z=-1\), but you need not work out the principal parts at that point.)
November 10
I posted solutions to the quiz, including a remark about Bernoulli numbers.
In class, we discussed the notions of zeroes and poles of functions.
Here is the assignment to hand in next time:
  • Exercise 4 on page 154 in section V.6.
  • Exercise 6 on page 158 in section V.7.
  • Exercise 1(f) on page 176 in section VI.2.
November 8
We discussed methods for computing power series (Taylor series, long division, undetermined coefficients, geometric series expansion), and we worked in groups on Exercises 1, 2, and 3 on page 153 in section V.6.
The assignment to hand in next time is a take-home quiz.
November 3
The second exam was given, and solutions are available.
November 1
We reviewed for the examination to be given on Thursday, and we discussed the principle that zeroes of analytic functions are isolated.
October 27
We discussed some previous exercises and saw that Taylor’s formula follows from Cauchy’s integral formula for a disk.
In view of the upcoming examination to be given on Thursday, November 3, the assignment (not to hand in) is to make a list of the three most important topics from each of Chapters III, IV, and V. The material that we have covered is
  • Sections 1–4 in Chapter III
  • Sections 1–5 in Chapter IV
  • Sections 3–4 in Chapter V.
October 25
We discussed infinite series of complex numbers, especially the ratio test and the root test, and we worked on some parts of Exercise 1 on page 143 in section V.3.
The assignment to hand in next time is parts (e), (f), (h), and (i) of Exercise 1 on page 143 in section V.3. There are answers in the back of the book, so the goal is more to understand the method than to find the answer.
October 20
We proved Liouville’s theorem about entire functions, the fundamental theorem of algebra, and the mean-value property of analytic functions on disks.
The assignment for next time is Exercises 2 and 3 on page 119 in section IV.5, which are applications of Liouville’s theorem.
October 18
We discussed Cauchy’s integral formula and the version for derivatives. Then we worked in groups on Exercise 1 on pages 116–117 in section IV.4. Parts (a), (b), (c), and (d) were submitted for a (group) quiz grade, and parts (e), (f), (g), and (h) are an individual homework assignment to hand in next time.
October 13
We used Cauchy’s theorem to prove that \[ \int_{0}^{\infty} \frac{x\sin (bx)}{x^{2}+1} \,dx = \frac{\pi}{2} e^{-b} \qquad \text{when \(b>0\)} \] by integrating over a rectangle and passing to the limit.
The assignment to hand in next time is Exercise 4 on page 106 in section IV.1 and Exercise 5 on page 113 in section IV.3.
October 11
We continued the discussion of the circle of ideas around exact and closed differentials, including the existence and computation of harmonic conjugate functions, Cauchy’s integral theorem, and the path-deformation principle.
The assignment to hand in next time is Exercises 2 and 3 on page 106 in section IV.1.
October 6
In connection with Green’s theorem, we discussed exact differentials, closed differentials, and path independence of integrals (three concepts that all coincide in simply connected domains in the plane).
The assignment to hand in next time is Exercises 1 and 2 on page 82 in section III.2 and Exercise 1 on page 106 in section IV.1.
October 4
I returned the graded exams, which were excellent. Congratulations!
As a follow-up, we looked at two interesting examples. The real-valued function \[ \begin{cases} \dfrac{x^2 y}{x^4+y^2}, & \text{if \( (x,y) \ne (0,0)\)}\\ 0, & \text{if \( (x,y) = (0,0)\)} \end{cases} \] has limit at \((0,0)\) equal to \(0\) along every line through the origin but has limit \(1/2\) along the parabola where \(y=x^2\), so the two-dimensional limit at the origin does not exist. The function \[ \begin{cases} \exp\left( \dfrac{-1}{z^4} \right), & \text{if \(z\ne 0\)}\\ 0, & \text{if \(z=0\)} \end{cases} \] satisfies the Cauchy–Riemann equations at the origin (because the \(x\) and \(y\) partial derivatives are both equal to \(0\)), but the function fails to have a complex derivative at the origin (because, for example, the directional derivative along the line where \(y=x\) does not exist).
Then we looked at the statement of Green’s theorem, and the proof for a rectangle. And we saw that a consequence of Green’s theorem is a version of Cauchy’s integral theorem: If a function is analytic in a region bounded by a simple closed curve, then the integral of the function around the boundary curve is equal to \(0\).
The assignment to hand in next time is Exercises 2 and 3 on page 75 in section III.1.
September 30
I posted the examination and solutions.
September 29
The first examination was given. I plan to post the examination and solutions tomorrow evening.
September 27
We discussed some additional properties of Möbius transformations (conformality, preservation of generalized circles, correspondence with matrix multiplication) and then reviewed for the examination on Chapters 1 and 2 that takes place next class.
September 22
We discussed Möbius transformations (linear fractional transformations): namely, one-to-one transformations of the extended complex numbers that arise by composing translations, real dilations, rotations, and inversion.
The assignment for next time (not to hand in) is to make a list of the four most important topics from each of Chapter I and Chapter II.
September 20
After we took a group quiz, we discussed the relationship between real-linear transformations of \(\R^2\) and complex-linear transformations of \(\C\), the relationship between the Jacobian matrix and the Cauchy–Riemann equations, and geometric interpretations of the complex derivative (local length magnification, local area magnification, and conformality).
The assignment for next time is Exercise I.6.2(a) on page 24, Exercise II.3.3 on page 50, and Exercise II.5.1(b) on page 57.
September 15
Since the first midterm examination is coming up on September 29 (two weeks from today), I added a link to some old Math 407 exams.
In class, we discussed the Cauchy–Riemann equations and some implications: namely, the real part (also the imaginary part) of an analytic function is a harmonic function (satisfies Laplace’s equation), and the level curves of the real part and the imaginary part of an analytic function form families of orthogonal trajectories.
The assignment for next time Exercises II.2.1(g) and II.2.2 on page 46 and Exercise II.3.5 on page 50.
September 13
We discussed a geometric solution to Exercise I.3.7 about stereographic projection; the definition of complex powers of complex numbers; and the definition of the complex derivative as a two-dimensional limit.
The assignment for next time is Exercise I.7.1(b) on page 27 and Exercise II.2.3 on page 46.
September 8
The assignment that was due today is deferred to next time. In class, we discussed some approaches to the first problem and had a preview of a coming attraction about convergence of infinite series of complex numbers. We also discussed the complex logarithm function and different possible domains for it.
There was a quiz, and solutions are available.
The assignment to hand in next time is the two problems originally due today plus Exercise 2 on page 31 in Section I.8.
September 6
We discussed geometric interpretations of some of the exercises in the first assignment. We determined all values of the complex variable \(z\) for which \(\sin(z)=4\). Generalizing, we saw that the complex sine and complex cosine functions have all of \(\C\) as range, and the complex exponential function has \(\C\setminus \{0\}\) as range.
The assignment for next time is to solve Exercise 5 on page 10 in Section I.2 and Exercise 7 on page 32 in Section I.8.
September 1
We discussed the complex numbers as an algebraically closed field that lacks an order relation but that does have a natural distance function (so we can talk about limits). We also discussed stereographic projection and the notion of the extended complex numbers equipped with a point at infinity. Additionally, we saw that if the exponential, sine, and cosine functions are defined by power series, then a direct consequence is Euler’s formula: \(e^{iz} = \cos(z)+i\sin(z)\).
We worked in groups on finding a geometric description (in terms of lines and circles) of the set of points \(z\) in the complex plane satisfying each of the following conditions:
  1. \(\Re(iz)=2\)
  2. \(|z-i| \ge 2\)
  3. \(z+\overline{z}=2\)
  4. \(\left| \dfrac{z-1}{z+1}\right| \le 2\)
The assignment for next time is to finish the preceding task and to solve Exercise 7 on page 15 in Section I.3 (about stereographic projection).
August 30
In class, we introduced ourselves and the complex numbers. As an example, we computed that \(\sqrt{3+4i} = \pm (2+i)\).
The assignment for next time is to start reading the textbook and to solve in Section I.1 Exercises 1c,d,e (page 4) and Exercise 6 (page 5).
August 28
I posted some errata for the textbook.
July 26, 2016
This site went live today. Once the Fall 2016 semester begins, there will be regular updates about assignments and the highlights of each class meeting.