General Information
| Meeting Time |
Mon., Wed., Fri., 9:00 - 9:50 |
| Location | Building 380 (Math), room 380-W
|
| Professor |
Soren Galatius (galatius@math.stanford.edu)
Office: 382-D (2nd floor, Math building 380)
Office Phone: 3-2794
Office Hours: Mondays,
Wednesdays, Fridays 1:30-2:30 or by appointment.
|
Course
Assistant |
Dmitriy Ivanov (dmivanov@stanford.edu)
Office: 380-U1 (Basement, Math building)
Office Hours:
Mondays: 4-5, Wednesdays: 2:30 - 3:30,
Tuesdays, Thursdays: 12:30 - 2:00, Fridays 12 - 1.
|
| Textbook |
Complex Variables and Applications, Seventh Edition, by J.W. Brown and R.V. Churchill. (problem numbers has changed since the sixth edition) |
Grade
Breakdown | Homework -- 20%, Midterms -- 20 % each, Final Exam -- 40% |
Course
Content | The course will first cover all or most of the first six chapters of Brown and Churchill's book. After that we will cover selected parts of the rest of the book. More details will appear in the syllabus below as we move along |
| Prerequisites |
Math 52.
|
| Exams |
1st Mid-term: Wed. October 20th (in-class).
2dn Mid-term: Wed. November 17th (in-class).
Final: Thu. December 9th (8:30 - 11:30) |
Syllabus
("B&C" is shorthand for the book by Brown and Churchill)
| Date |
Material covered |
Relevant Reading |
| Mon. 9/27 |
Complex numbers and their algebraic
properties. I spent the last 15 minutes trying to give a
brief idea about the rest of the course, which will be
about functions from C to C. |
B&C, Ch. 1 |
| Wed. 9/29 |
Modulus and argument of a complex
number. The argument is multivalued. Exponential form of a
complex number. Multiplication and division in exponential
form. Powers. n'th roots of complex numbers. A complex
number (different from 0) has exactly n different n'th root. |
B&C, Ch. 1 |
| Fri. 10/1 |
Roots of unity. Subsets of the complex
plane: interior, exterior and boundary points. Open subsets
and closed subsets. |
B&C, Chs. 1 |
| Mon., 10/4 |
More on subsets: The closure of a
subset S of C. Connected open subsets (=domains). Complex
functions. Analytically these can be represented as f(z) =
u(x,y) + i v(x,y) or as f(z) = u(r,theta) + i v(r,theta).
Geometrically we can't draw their graphs, but we can think of
them as "moving points in C around". When we think of a
function geometrically, we sometimes call it a mapping,
although logically it is the same as a function. We defined
the very important exponential function. Finally we discussed
limits. The last theorem said that a complex function has a
limit as z -> z0 if and only if both the real parts and the
imaginary parts have limits. |
B&C, Ch. 1 and 2 |
| Wed., 10/6 |
I finished the paragraph "theorems on
limits", skipping some of the proofs. The proofs are just as
in the real case, and are technically very similar to the
proofs I did in paragraph 17. We skipped paragraph 16 for now
(maybe we'll return to it later). Finally we proved the two
theorems in paragraph 17. |
B&C Ch. 2 |
| Fri., 10/8 |
Derivatives, differentiation formulas. Cauchy-Riemann equations |
B&C Ch. 2 |
| Mon., 10/11 |
We finished the proof that f is (complex)
differentiable if u and v are (real)
differentiable and satisfy the Cauchy-Riemann
equations. We noted that the Cauchy-Riemann
equations are equivalent to the statement "the
gradient of v is the vector obtained by rotating
the gradient of u pi/2 counterclockwise". Then
using the formula for the gradient in polar
coordinates we derived the Cauchy-Riemann
equations in polar coordinates. Analytic
functions and their singularities. Entire
functions. Finally we defined what a harmonic
function is. |
B&C, Ch. 2 |
| Wed., 10/13 |
Harmonic functions: main theorem was that
the real and imaginary parts of an analytic
function were both harmonic. Harmonic
conjugates: f is analytic if and only if v is a
harmonic conjugate of u. Then we discussed the
question of whether any function has a harmonic
conjugate (i.e. is the real part of an analytic
function). This is true locally, but not
globally. We wrote the Cauchy-Riemann equations
as grad(v) = (-u_y, u_x) and noted that u being
harmonic meant precisely that the vectorfield
(-u_y, u_x) had curl zero. |
B&C Ch. 2 |
| Fri., 10/15 |
Uniquely determined analytic functions.
The main theorem (from the book) is that an
analytic function is uniquely determined by its
values on a line in a domain. I gave several
applications of this theorem (not in the book):
The definitions of exp, cos, sin, ... are the
"right" definitions. Also "formulas" that hold
on the real line can be extended to C using the
uniqueness theorem. After that we went to
chapter 3 and discussed the exponential function
and the logarithm.
|
B&C Ch. 2 and 3 |
| Mon., 10/18 |
More on logarithms. Branches of the
logarithm. The derivative of (a branch of) the
logarithm. Complex exponents. I gave the
example 1^(i/pi) to show that these are
ill-behaved. Finally I talked about one of the
homework problems. |
B&C Ch. 3 |
| Wed., 10/20 |
Exam I (in-class). Open book,
open notes. Last year's exam. Sample solution to problem 2. |
B&C Ch. 1 and 2 |
| Fri., 10/22 |
Trigonometric functions. Sections 34 and
35 on hyperbolic and inverse trig functions were
be skipped. The remaining time was spent on
Wednesday's exam. |
B&C Ch. 3 |
| Mon., 10/25 |
Defined integrals and derivatives of a
complex function w(t) of a real variable t.
This was just defined by
differentiating/integrating the real and
imaginary parts separately. Then I defined the
notion of arcs and contours. |
B&C Ch. 4 |
| Wed., 10/27 |
Given a continuous complex function f of a
complex variable and a contour C in the domain
of definition of f, you can integrate f over C.
I defined this using the concepts from Monday's
lecture. Then I did three examples. Two of the
examples were example 1 and 3 in section 40 in the
book. The last one was integration of 1/z along a
contour starting at 1 and ending at -1. We saw
that the result depended on the chosen contour,
even though 1/z is analytic. Finally, I defined
lengths of contours and proved the upper bound (1)
on page 131 in the book. |
B&C Ch. 4 |
| Fri., 10/29 |
Antiderivatives. |
B&C Ch. 4 |
| Mon., 11/1 |
The Cauchy-Goursat theorem. Simply
connected domains. |
B&C Ch. 4 |
| Wed., 11/3 |
Proved the Cauchy integral formula. |
B&C Ch. 4 |
| Fri., 11/5 |
Used Cauchy's integral formula to prove
that all derivatives of an analytic function
exist. We did that by differentiating the
Cauchy integral formula, and in fact we found a
formula for the n'th derivative. |
B&C Ch. 4 |
| Mon., 11/8 |
Used the Cauchy integral formula to prove
the Cauchy inequalities. Liouville's theorem.
Used Liouville's theorem to prove the
"fundamental theorem of algebra": every
polynomial has a root. In fact we proved that
every polynomial can be written as
(z-z1)(z-z2)...(z-zn) times a constant. |
B&C Ch. 4 |
| Wed., 11/10 |
Finished the proof of the maximum modulus
theorem. It says that if f is analytic on D, then
the real function |f(z)| (the modulus of f) does
not have a maximum in the domain. This finishes
chapter 4. |
B&C Ch. 4 |
| Fri., 11/12 |
Series |
B&C Ch. 5 |
| Mon., 11/15 |
Series |
B&C Ch. 5 |
| Wed., 11/17 |
Exam II (in-class). Open book,
open notes. Last year's exam.
My solutions: Page 1,
Page 2, Page 3, Page 4 |
B&C Ch. 1, 2, 3 and 4. |
| Fri., 11/19 |
Series |
B&C Ch. 5 |
| Mon., 11/22 |
Series. The residue theorem. Midterms were given back. |
B&C Ch. 6 |
| Wed., 11/24 |
The residue theorem: poles, essential singularities. Methods to compute residues at poles. Example. |
B&C Ch. 6&7 |
| Fri., 11/26 |
Thanksgiving. No lecture. My solutions to the
midterm is here: Page 1,
Page 2, Page 3, Page 4 |
B&C Ch. 5 |
| Mon., 11/29 |
The residue theorem -- more examples |
B&C Ch. 7 |
| Wed., 12/1 |
Rouche's theorem |
B&C Ch. 7 |
| Fri., 12/3 |
More examples. My solutions to some of the homework problems: page 1, page 2, page 3, page 4, page 5, page 6, page 7. |
B&C Ch. 7 |
| Thu., 12/6 |
Final. 8:30 - 11:30. Room 380-W. Last years exam. |
B&C Ch. 1-7 |
Announcements & Dates
- Important Dates and Class Holidays
- Sunday, October 17th: Add deadline
- Wednesday, October 20: Exam 1 (In-class)
- Sunday, October 24th: Drop deadline
- Wednesday, November 17: Exam 2 (In-class)
- Friday, Nov 26: Thanksgiving
- Thursday, December 9: Final Exam, 8:30-11:30 AM, (Location to be determined)
- No announcements yet...
Homework
Homework assignments are due in class each Friday
(except for Friday, Oct 1st), unless otherwise noted. You may discuss homework problems with
each other, but you must write it on your own. As usual, a
correct answer is not enough to recieve full credit -- you must
also show your reasoning. Also you must write when and what
theorems/equations from the book you are using.
|
Assignment |
Due date |
| #1 |
B&C: p11 #4, p13 #2, p21 #1,10, p28 #2,6,
p31 #1,4, p35 # 2,4, p42 # 2,7 |
Due: Friday, October 8th
|
| #2 |
B&C: p53 #5,9, p59 #1,3,8(a), p68 #1(c,d),3(b),7, p73 #1(b,c),6 |
Due: Friday, October 15th
|
| #3 |
B&C: p73 #7, p78 #1(b,d),4 p89 #7,8(b),
p94 #5(b),6, p99 #1(b)
| Due: Friday, October 22th
|
| #4 |
B&C: p94 #10, p99 #2(c) p103 #1,5,16,18
p115 #4,5
| Due: Friday, October 29th
|
| #5 |
B&C: p115 #7, p129 #1(c),3,6,10,11 p133 #1,2,6
p141 #3
| Due: Friday, November 5th
|
| #6 |
B&C: p153#1(c,f),2,7, p162#1,2, p171#1,2
| Due: Friday, November 12th
|
| #7 |
B&C: p188#3,5(a),6,10, p198#4,5, p212#1,4
| Due: Wednesday, November 24th
|
| #8 |
B&C: p233#1(a,b),2(b), p238#1(c),3, p257#3,4,7, p265#1,4
| Due: Wednesday, December 1st
|
|
