Record of daily activities and homework,
Math 617, Theory of Functions of a Complex Variable I,
Fall 2003
- Monday, September 1
- We discussed three definitions of the complex
numbers: the everyday working definition as the set of
a+bi with the usual algebraic operations
subject to the rule that
i2=-1;
the formal definition as the set of ordered pairs of
real numbers with componentwise addition and a special
multiplication rule; and the algebraic definition as
R[x]/(x2+1). Also
we looked at the condition for a real-linear
transformation of R2 to correspond to
a complex-linear mapping of C.
- Homework for Wednesday: Read sections 1.1
to 1.3 on pages 1-14 and do exercises 2b, 3a, 13a, and
14c on pages 20-22.
- Wednesday, September 3
- We discussed the notion of continuity for functions
from C to C and also the notion of
complex differentiability (Cauchy-Riemann equations).
- Homework for Friday: Read sections 1.4 and
1.5 on pages 14-20 and do exercises 8, 9, 18, and 33 on
pages 21-23.
- Friday, September 5
- We continued the discussion of the Cauchy-Riemann
equations as the defining property of holomorphic
functions. In particular, we interpreted the
Cauchy-Riemann equations for f(z) as being the standard
necessary condition from advanced calculus
for exactness of the differential f dz.
- Homework for Monday: Read section 2.1 and
the first part of section 2.2, pages 29-37, and do
exercises 34, 37, and 47 on pages 25-26
(Chapter 1) and exercise 1 on page 60
(Chapter 2).
- Monday, September 8
- We discussed the definition of the logarithm as a
holomorphic function, and we worked some concrete
examples of computing line integrals,
- Homework for Wednesday: Read the end of
section 2.2, all of section 2.3, and the beginning
of section 2.4, pages 38-45, and do exercises 3,
5, and 8 on pages 60-61 (Chapter 2) and
exercise 52 on page 27 (Chapter 1).
- Wednesday, September 10
- We discussed Cauchy's Integral Theorem from the
point of view of closed and exact forms and Green's
Theorem.
- Homework for Friday: Read pages 46-52 (the
end of section 2.4 and section 2.5). Do
exercise 55 on page 27 (Chapter 1) and
exercises 9b, 15, and 17 on pages 61-62
(Chapter 2).
- Friday, September 12
- We discussed the various different
characterizations that we know so far of holomorphic functions.
- Homework for Monday: Read
section 2.6, pages 53-60, and the start of
section 3.1, pages 69-72. Do exercises 18c, 23,
and 36 on pages 62-66 (Chapter 2).
- Monday, September 15
- We discussed the significance of the Cauchy
integral formula and looked at examples of using it to
compute integrals.
- Homework for Wednesday: Read the proof of
Morera's theorem on pages 73-74 and section 3.2 on
pages 74-81. Group z should do starred problem
4z on page 67 to present in class.
- Wednesday, September 17
- We looked at problems 41, 42, and 44 on
page 67 at the end of Chapter 2 related to
Morera's theorem and special path integrals.
- Homework for Friday: Read section 3.3
(pages 81-84) and do exercises 9, 10, and 11(a) on
page 95 (Chapter 3).
- Friday, September 19
- We discussed convergence of power series in the
complex domain.
- Homework for Monday: Read section 3.4
(pages 85-88) and do exercises 19, 20(a), and 26
on pages 96-97 (Chapter 3).
- Monday, September 22
- We worked an exercise on
uniform convergence.
- Homework for Wednesday: Read sections 3.5
and 3.6 (pages 88-94) and do exercises 15, 17,
and 21 on pages 96-97 (Chapter 3).
- Wednesday, September 24
- We discussed some of the homework exercises, and we
took a quiz.
- Homework for Friday: The new groups will
prepare exercises 44, 45, 46, and 48 on pages
100-101 (Chapter 3) for presentation in class.
- Friday, September 26
- We discussed the four problems assigned on
Wednesday. The solution of problem 46 involved the
Arzelà-Ascoli theorem, for which one may find the
statement and a reference on page 487 in
Appendix A of the textbook.
-
Homework for Monday: Study for the
examination.
- Monday, September 29
- First examination
- Wednesday, October 1
- The graded examinations were returned, and we
discussed the solutions. Also we discussed the tripartite
classification of isolated singularities as removable singularities,
poles, or essential singularities.
-
Homework for Friday: Read section 4.1
(pages 105-109) and do exercises 3, 5(c,d), and 7
on pages 145-146 (Chapter 4).
- Friday, October 3
- We discussed Laurent series.
-
Homework for Monday: Read section 4.2
(pages 109-113) and section 4.4 (pages 119-122)
and do exercises 4, 13(b), and 17(b) on pages 145-148
(Chapter 4).
- Monday, October 6
- We worked on exercises 2, 6, 9, and 12 in
Chapter 4 (pages 145-147).
- Homework for Wednesday: Read section 4.3
(pages 113-119). Each group should be prepared to
present the solution to its problem on Wednesday.
- Wednesday, October 8
- We looked at solutions to the problems from last
time.
- Homework for Friday: Read
section 4.5 (pages 122-127) and do exercises 14,
23, and 36(a) on pages 147-150 (Chapter 4).
- Friday, October 10
- We worked on exercises 35 and 36 on
page 150 (calculation of residues and
integrals).
- Homework for Monday: The groups will
prepare the following items. (A) Find and correct
the mistake in Example 4.6.3 (group 2);
(B) Fill in the details of Example 4.6.6 (group 3);
(C) Exercise 54, page 154 (group 1);
(D) Exercise 57, page 155 (group 4).
- Monday, October 13
- We discussed solutions to problems 54
and 57.
- Homework for Wednesday: Read
section 4.6 (pages 128-137). The groups will
prepare the following items. (A) Exercise 53 on
page 154 (group 1); (B) Exercise 58
on page 155 (group 2);
(C) Exercise 61 on page 155
(group 3); (D) Use contour integration to
prove that the sum of 1/k4 over
positive integers k equals
π4/90 (group 4).
- Wednesday, October 15
- We discussed Examples 4.6.3 and 4.6.6 of
contour integration.
- Homework for Friday: Read section 4.7
(pages 137-145) and do exercises 37 and 40 on
pages 151-152 (Chapter 4).
- Friday, October 17
- We discussed meromorphic functions,
Mittag-Leffler's theorem, singularities at infinity,
and the rational parametrization of the unit
circle.
- Homework for Monday: Do exercises 42, 43,
and 44 on pages 152-153 (Chapter 4).
- Monday, October 20
- We discussed the computation by residues of the sum
of 1/k4 over the positive
integers k,
the Riemann zeta function,
and related topics.
- Homework for Wednesday: Read
section 5.1, pages 157-162, and review for the
take-home examination to be distributed at the end of
class on Wednesday.
- Wednesday, October 22
- We discussed the geometric viewpoint on inversion
in the complex plane and on stereographic
projection. The take-home
examination was distributed.
- Friday, October 24
- No class meeting because of the take-home
examination that is due at the beginning of class
on Monday.
- Monday, October 27
- The discussion was an overview of the key topics in
Chapter 5: the argument principle, the open
mapping theorem, the maximum principle, the Schwarz
lemma, the group of holomorphic automorphisms of the
unit disc, Hurwitz's theorem, and Rouché's
theorem.
- Homework for Wednesday: Read sections 5.2
and 5.3 (pages 162-169) and do exercises 2, 3,
and 6 on pages 174-175 (Chapter 5).
- Wednesday, October 29
- We worked on problem 10 on page 176
(Chapter 5) having to do with Rouché's
theorem.
- Homework for Friday: Read sections 5.4
and 5.5 (pages 169-174) and do exercises 5, 10(f),
and 16 on pages 174-177 (Chapter 5).
- Friday, October 31
- We discussed some exercises involving
Rouché's theorem (in particular, the
interpretation of the hypothesis as meaning that
equality does not hold in the triangle inequality).
Also, we looked at a concrete solution of the fourth
problem on the second
examination.
- Homework for Monday: Do exercises 11
and 14 on pages 176-177 (Chapter 5).
- Monday, November 3
- We worked on an exercise on properties of
holomorphic functions.
- Homework for Wednesday: Read sections
6.1-6.3 (pages 179-189) and do exercise 8 on
page 202 (Chapter 6).
- Wednesday, November 5
- We discussed properties of linear fractional
transformations and their representation on projective
space.
- Homework for Friday: Read section 6.4
(pages 189-192) and do exercises 1 and 27 on pages
202-205 (Chapter 6).
- Friday, November 7
- We continued the discussion of linear fractional
transformations and of quantities that they preserve
(cross ratios, symmetry, angles).
- Homework for Monday: Read section 6.5
(pages 192-196) and do exercises 20 and 32 on
pages 204-206 (Chapter 6).
- Monday, November 10
- We worked on an exercise on normal families of
holomorphic functions.
- Homework for Wednesday: Read sections
6.6-6.7 (pages 196-201). Groups 1, 2, 3, and 4
will do exercises 21, 22, 23, and 24 respectively
(pages 204-205, Chapter 6).
- Wednesday, November 12
- We discussed solutions to exercises 21 and 22
on page 204 (Chapter 6).
- Homework for Friday: Read sections 7.1
and 7.2 (pages 207-212) and do exercises 4, 10,
and 17 on pages 243-245 (Chapter 7).
- Friday, November 14
- We worked on an exercise on properties of harmonic
functions.
- Homework for Monday: Read section 7.3
(pages 212-218) and do exercises 11 and 15 on
pages 244-245 (Chapter 7).
- Monday, November 17
- We discussed the Poisson integral and compared its
properties to those of the Cauchy integral. Also we
summarized properties equivalent to simple connectivity
of planar regions.
- Homework for Wednesday: Read
section 7.4 (pages 218-220) and review for the
examination to be given on Friday.
- Wednesday, November 19
- We reviewed for the examination on Chapters 5
and 6 to be given on
Friday.
- Homework for Friday: Prepare for the
examination.
- Friday, November 21
- Third examination
- Monday, November 24
- We discussed solutions to the third
examination.
- Wednesday, November 26
- There will be no class meeting this day. Enjoy the
Thanksgiving holiday. The next class meeting will be
Monday, December 1.
- Monday, December 1
- We discussed subharmonic functions and Perron's
method for solving the Dirichlet problem.
- Homework for Wednesday: Read
section 7.7 (pages 227-236) and do exercises 43
and 49 on pages 247-248 (Chapter 7).
- Wednesday, December 3
- We worked on an exercise on subharmonic functions
and Perron's method.
- Homework for Friday: Read section 7.6
(pages 224-226) and section 7.8 (pages 236-239).
The groups will prepare the remaining parts of the
class exercise on subharmonic
functions and exercise 30 on page 246
(Chapter 7).
- Friday, December 5
- We discussed solutions to the exercise on subharmonic
functions from last time and also to
exercise 30 on page 246
(Chapter 7).
- Homework for Monday: Read
section 7.5 (pages 220-224) and section 7.9
(pages 240-243).
- Monday, December 8
- This was the last class meeting for the semester.
We did the course evaluations and discussed the Schwarz
reflection principle and holomorphic mappings of
annuli. The final examination is scheduled for Tuesday,
December 16 from 10:30 to 12:30.