# Math 407, Section 500, Complex Variables, Spring 2008

Harold P.
Boas

- Thursday, May 8
- You can check your grades for this course at the TAMU eLearning site. Have a great summer!
- Wednesday, May 7
- The final exam was given, and solutions are available. The median score was an impressive 84. Good work!
- Tuesday, April 29
- At our final class meeting, we discussed the Riemann hypothesis, one of the major open problems in mathematics.
- Monday, April 28
- We listed some of the main topics of the semester and reviewed for the comprehensive final exam to be given on Wednesday, May 7, from 10:30 to 12:30.
- Friday, April 25
- We discussed the fourth problem on the quiz, for which solutions now are available; the Schwarz-Christoffel formula for mapping the upper half plane onto a polygonal region; and the example of mapping the unit disc to the unit disc with a slit in it.

The assignment is to compile a list of the ten most important topics/concepts/theorems from the course. - Wednesday, April 23
- We attempted to show that the Schwarz-Christoffel transformation w=∫
_{0}^{z}{t(1-t^{2})}^{-1/2}dt maps the upper half plane to a square with sides of length ∫_{0}^{1}{t(1-t^{2})}^{-1/2}dt. The eleventh quiz was distributed as an open-book take-home quiz due next class. - Monday, April 21
- By way of motivating the Schwarz-Christoffel formula, we discussed the conformal mapping of a half-strip onto the upper half plane by the sine function and how one might independently arrive at a formula for the inverse mapping by thinking about what form the derivative must have.

The assignment is to elaborate on exercise 5 on page 241 in section 3.5 as follows (taking R=1 for simplicity). (a) Verify that f(z)=(z+z^{-1})/2 is a correct answer. [Hint: see Example 5 on page 211 and Exercise 5 on page 218.] (b) Show that another correct answer is g(z)=-(1/4)(z+1)^{2}/(z-1)^{2}. [Hint: Use the preliminary transformation 1/(z-1) to map to a quadrant; use a translation and a rotation to get to the first quadrant; then use a square mapping to get to the upper half plane.] (c) Find a linear fractional transformation T(z) such that g(z)=T(f(z)). - Friday, April 18
- We worked in groups on exercises 3-9 (section 3.5) about conformal mapping.

The assignment is to begin reviewing for the comprehensive final exam. We will have an open-book review quiz sometime next week. - Wednesday, April 16
- We looked at some concrete examples of conformal maps with the assistance of Maple to plot the orthogonal systems of image curves. The tenth quiz was given, and solutions are available.

The assignment is to read the beginning of section 3.5, pages 224-227, and to convince yourself that you understand the pictures on pages 226-227. - Monday, April 14
- We further discussed fixed points of analytic automorphisms of the disc, we reviewed the classification of isolated singularities, and we stated the Casorati-Weierstrass theorem and the great Picard theorem.

The assignment is to finish the previous homework, if you have not already done so, and to do exercise 8 on page 218 in section 3.4 and exercise 2 on page 241 in section 3.5. - Friday, April 11
- We discussed some topics related to the quiz problems from last time: in particular, approaches to the third problem using symmetry and conformality. Then we went off on a tangent about the Poincaré disc model of non-Euclidean geometry (not in the textbook).

The assignment for Monday is as stated below. - Wednesday, April 9
- We discussed the Riemann mapping theorem. Then we worked in groups for a quiz grade on exercise 16 on page 85 in section 2.1, exercise 26 on page 152 in section 2.5, and exercise 10 on page 218 in section 3.4; we did not finish, so the assignment for Friday is to complete these exercises.

For Monday, the assignment is to do exercises 1, 3, and 5 on page 218 in section 3.4. (Since these exercises have sketches of solutions in the back of the textbook, you need to fill in the details.) - Monday, April 7
- We discussed the preservation of symmetry by Möbius transformations and the property of conformality of analytic mappings with nonzero derivative.

The assignment for next time is to read section 3.4 and to be prepared for a quiz. - Friday, April 4
- We discussed one of the homework problems and some related topics such as the conformality of linear fractional transformations, the representation of Möbius transformations by matrices, and the orientation-preserving property of analytic mappings.

The assignment for next time is to do exercise 28 on page 56 in section 1.5, exercise 20 on page 207 in section 3.3, and exercise 15 on page 219 in section 3.4. - Wednesday, April 2
- We discussed the theorem that Möbius transformations take generalized circles to generalized circles. The eighth quiz was given, and solutions are available.

The assignment for next time is to do exercises 7a,c and 8a on page 205 in section 3.3 and 38 on page 22 in section 1.2. - Monday, March 31
- The graded exams were returned. The class median was 82, and there were two scores of 105. Good job!

In class, we discussed Möbius transformations (linear fractional transformations). The assignment for next time is to read pages 196-201 in section 3.3 and to do exercises 4c,d and 5a on page 204. - Friday, March 28
- The second examination was given, and solutions are available.
- Wednesday, March 26
- By way of review for the exam to be given on Friday, we discussed the argument principle, the maximum-modulus principle, and the local geometric behavior of analytic functions.
- Monday, March 24
- Reminder: the second examination (covering sections 2.3-2.6, 3.1, and 3.2) will be given on Friday.

In class, we worked in groups on exercise 13 on page 180 in section 3.1 (about Rouché's theorem) and exercise 3 on page 194 in section 3.2 (about the maximum-modulus principle) for a quiz grade.

The assignment for next time is to compile a list of the main concepts from sections 2.3-2.6, 3.1, and 3.2. - Wednesday, March 19
- We worked on exercise 13 from page 167, an integral using the keyhole contour.

Friday, March 21, is a reading day; your reading assignment is section 3.1 (pages 171-179) and section 3.2 (pages 191-194).

Reminder: the second examination will be given on Friday, March 28. - Monday, March 17
- We used contour integration to compute the real integrals ∫
_{0}^{∞}(1+x^{2})^{-1}dx and ∫_{0}^{∞}cos(x)/(1+x^{2}) dx.

The assignment for next time is to do exercises 3 and 5 on page 167 in section 2.6. Notice that the answers are in the back of the textbook. - Friday, March 7
- We used the Residue Theorem to compute the real integral
∫
_{0}^{∞}(1+x^{4})^{-1}dx. Also, we watched the prize-winning video "Möbius Transformations Revealed" by mathematicians Douglas Arnold and Jonathan Rogness.

The assignment is to have a good spring break. - Wednesday, March 5
- We continued the discussion of residues. The sixth quiz was given, and solutions are available.

The assignment for next time is to attempt one of the integrals on page 167 at the end of section 2.6 (not to hand in). - Monday, March 3
- We discussed residues and Laurent series.

The assignment for next time is to do exercises 6, 8, and 12 on page 150 in section 2.5. - Friday, February 29
- We discussed solutions to the homework exercises. The fifth quiz was given, and solutions are available.

The assignment is to read pages 141-147 in section 2.5 and to try exercises 7 and 13 on page 150 (not to hand in). - Wednesday, February 27
- We discussed the notions of the order of a zero and the order of a pole of an analytic function.

The assignment for next time is to read the first part of section 2.5, pages 135-141, and to do exercise 6 on page 116 in section 2.3 (the answer is 2π divided by √7), exercise 20 on page 133 in section 2.4, and exercise 2 on page 150 in section 2.5. - Monday, February 25
- We discussed some applications of Cauchy's integral formula and worked on some exercises from the end of section 2.4.

The assignment for next time is to do exercises 8, 16, and 18 on page 133 in section 2.4. - Friday, February 22
- We discussed the existence of anti-derivatives of analytic functions in simply connected domains and also the homework exercises about the mean-value property and the maximum principle. The fourth quiz was given, and solutions are available.

The assignment for next time is to read section 2.4. - Wednesday, February 20
- We continued the discussion of Cauchy's theorem and Cauchy's integral formula, touching on the principle of deformation of curves and the question of path independence.

The assignment for next time is to do exercises 12, 14, and 16 on page 117 in section 2.3. Be prepared for a quiz on section 2.3. - Monday, February 18
- The graded exams were returned. The class median was 82, and there were two scores of 100. Good job!

In class, we discussed Cauchy's theorem and Cauchy's integral formula.

The assignment for next time is to read section 2.3 and to do exercises 2, 4, and 8 on page 116 in section 2.3. - Friday, February 15
- The first examination was given, and solutions are available.
- Wednesday, February 13
- We reviewed for the examination to be given next class over Chapter 1 and sections 2.1 and 2.2.
- Monday, February 11
- We worked on some exercises from section 2.1 about the Cauchy-Riemann equations and about finding harmonic conjugates.

The assignment for next time, by way of review for the exam to be given on Friday, is to make a list of the main concepts and theorems that we have covered (not to hand in). - Friday, February 8
- We discussed convergence of power series (of a complex variable), the radius of convergence, the ratio test, and the root test.

The assignment for next time is to read section 2.2.

Reminder: the first examination will be next Friday, February 15, on Chapter 1 and sections 2.1 and 2.2. - Wednesday, February 6
- We discussed the definition of the complex derivative, examples of differentiable and non-differentiable functions, and the Cauchy-Riemann equations. The third quiz was given, and solutions are available.

The assignment for next time is to do exercise 16 on page 75 in section 1.6 and exercises 4 and 10 on pages 84-85 in section 2.1. - Monday, February 4
- We discussed Green's theorem in complex form, the definition of analytic functions, and the path independence of line integrals of analytic functions.

The assignment for next time is to do exercises 4, 6, and 14 on pages 73-74 in section 1.6 and to read section 2.1. - Friday, February 1
- We discussed line integrals and Green's theorem. The assignment for next time is to read section 1.6.
- Wednesday, January 30
- We studied the function sin(z) as a geometric mapping. The second quiz was given, and solutions are available.

The assignment to hand in next time is: (a) exercises 16 and 20 on page 54 in section 1.5; and (b) determine the image under the function w=cos(z) of the half strip in the z-plane where y>0 and 0<x<π. - Monday, January 28
- We discussed the complex exponential and logarithm functions.

The assignment for next time is to do exercises 2, 4, 8, and 14 on page 53 in section 1.5 (to hand in). Notice that there are infinitely many values for exercise 4, because lowercase "log" means all possible branches of the logarithm function. - Friday, January 25
- We discussed some of the homework problems, and we looked at some examples of infinite series of complex numbers, particularly ∑
_{n≥1}i^{n}/n. Incidentally, the exact sum of this conditionally convergent series is -(ln√2)+i(π/4), and we shall see why later in the course.

The assignment for next time is to read section 1.5 about the exponential, logarithm, and trigonometric functions. - Wednesday, January 23
- We discussed the geometry of inversion and looked at some examples of the topological notions of boundary and closed set.

The assignment for next time is to finish reading section 1.4 and to do the following exercises to hand in:- number 18 on page 10 (section 1.1)
- number 38 on page 22 (section 1.2)
- number 10 on page 28 (section 1.3)

- Friday, January 18
- We discussed the calculation of roots in the complex plane before turning to the topological notions of open set, closed set, and boundary. The first quiz was given, and solutions are available.

The assignment for next time is to read section 1.3 and the first three pages of section 1.4, and do the following exercises to hand in: numbers 22 and 26 on page 21; number 36 on page 22; and number 8 on page 28. - Wednesday, January 16
- We discussed the derivation of the Cauchy-Schwarz inequality (exercise 21 on page 10 of the textbook), and we worked in groups on some exercises from section 1.2 (page 20 of the textbook).

The assignment for next class is to read section 1.2 and to write up solutions to hand in for exercise 15 on page 9, exercise 16 on page 10, and exercise 18 on page 21. Notice that exercise 9 has a sketch of a solution in the back of the book, so you should fill in the details to provide a complete solution. - Monday, January 14
- This site went live today. The
first-day handout is available online.

The assignment for next class is to read section 1.1 and to do exercises 5a, 7, 13a, 19, and 21 on page 9. (The answers to odd-numbered exercises are in the back of the book, so this first assignment is not to be turned in.) Be prepared for a short quiz next class on section 1.1.

These pages are copyright © 2008 by Harold P. Boas. All rights
reserved.