MAT 412 and 412ZZ Complex Analysis for Applications
ES 143 MWF
Instructor: Professor Edward
Thomas ES 132F phone 442-4623
Best way to reach me: et392@albany.edu
Text: Fundamentals of Complex Analysis ( 3rd edition) by Saff
and Snider.
The core material of the course is found in
Chapters 1 through 6. We will focus on computation ( rather
than theory) and on the applications of complex analysis to problems in physics
and engineering.
I’m going to assign daily problem sets. You
will complete them and hand them in at the next class. I’ll grade them and
get them
back to you pronto. They will count one third of the final grade. There will
also be two one hour exams, each
covering
about a half of the course and counting a third of the final grade.
Just for emphasis...when I say that the assigned
problems will count 1/3 of the grade, I’m not kidding.
There
are some problems that take so much time that they MUST be done outside class.
We’ll go over each
assignment
in class and you will have the opportunity to correct it, before handing it in.
But, I don’t want to grade batches of old homework. If you can’t get an
assignment done for some reason, get in touch with me in a timely way.
****************************
When you sign on to this course you are
paying for me to be your guide through some rough terrain and, in that
capacity,
I
have a couple of rules.
>First,
NO CELL PHONES or other electronic devices will be used while I am teaching.
TURN THEM OFF
>Second, please be in class, ready to go,
when class starts at
come late
to class and WE DEFINITELY WANT TO AVOID
THAT.
>And,
lastly, classes start on Monday, August 31st. In the past, some
people imagined that they could start classes when they chose.
That
proved to be a MISTAKE.
*******************************************
ASSIGNMENTS
1) pg 5 # 5,6,7,8,9
2) pg 5 #16, 17 and pg 13 # 7
and pg 23 # 6 a,b and 7 a,b,c,d
3) page 31 # 1, 2 , 3, 4 On 3c, it
helps to work from the inside out. First compute exp(i) using Euler’s
Formula. You’ll get cos and sin of 1 radian coming in
at this point. Then compute exp of that.
4) Compute the integral from
0 to 2Pi of cos(theta)^6 as discussed in class. Plus pg
31 # 12 both parts.
5) Pg 37 # 5 b, d, e and
6a plus: derive the formula for the
integral from 0 to 2Pi of
cos(theta) raised to any positive even
power, 2n.
6) page
56 # 3 and #4
7) a) Write these functions in u + iv form: f(z) = z^3, f(z) = exp(z) , f(z) = 1/z^2
b) Check the Cauchy Riemann Equations for f(z) = z^3, f(z) = exp(z), f(z) = 1/z ( just 1/z, not 1/z^2)
8) a)
Using the CR equations, show that the following are not differentiable: f(z)
=the conjugate of z , f(z) = the modulus of z and f(z)= ( x^2 +y^2) +i( 2xy)
b) Check to see if the CR
equations hold and if so find the derivative of f:
f(x,y) = x - 2xy +i(y –y^2
+x^2) and f(x,y)
= (x^2 + x + y^2)/ ( x^2+y^2) +i/(x^2+y^2)
Bonus: See if you can write these in terms of
z.
9) pg 84 # 3 b, c, e Bonus: If v is the harmonic conjugate of u,
then uv is harmonic. ( Added on Tuesday morning...did you know that -arctan(x/y) = arctan ( y/x) + a constant?
This might help on part e.)
10) page
85 # 9 and #11
11) page
171 # 3, 5, 10
12) pg 178 # 1 a through e plus finish the proof
of Cauchy’s Theorem as started in class. Added Tuesday AM...
Omit parts c and d of problem #1
13) page 201 #9 and page
203 #13, 17, 20 done by Cauchy’s Integral Formula.
14) page 212 # 3 d, e, f
and #4
15) page
123 #1 and 5a
16) Finish the Dirichlet problem that we started in class.
17) Compute the Maclaurin
series for cosz and sinz by
systematically calculating derivatives. Plus on page 250 #5 a, b, c, d
18) page 259 # 3 plus
find the Maclaurin series and the disk of convergence
for f(z) = z/(1-3z) and for f(z) = 3/(2-z)
19) page
276 # 1 a, b and #3 a, b, c
20) page
276 # 1 c, d plus # 5 plus #7 a
Sunday morning....On #5, I found it helpful to write (z+1)/z as 1 + 1/z
and then write that z as (z-4) +4 and factor out a 4. On 7a, multiply the
series for exp(1/z) times the series for 1/( 1-z^2).
For the latter series, you need to factor out a z^2.
21) page 285 #1 a
through e Tuesday morning: We didn’t have time to look at everything on
my list, so let me change the
assignment. Just do parts a, b and c. In that regard, let me direct your
attention to Lemma 7 on page 281. That is a HUGE help since it avoids expanding
out Laurent series.On Wednesday we’ll fill in the
rest of the details and finish this section in an orderly fashion, hopefully
without the dentist drill in the background.
22) page
285 # 1 d, g plus page 313 # 1 a –e
23) page 313 # 3 a, b,
d, e
24) page
317 # 1, 2, 3 Some peeps are having
problems with #3 simply because the algebra gets complicated. If you play your
cards right, you come down to 1/2i times the integral of z dz
divided by (z^2+3z+1)^2. DO NOT expand out the
denominator. Instead locate the zeros of that quadratic, factor it into two
linear factors each squared. This makes it MUCH easier to calculate the residue
at the pole that lies inside the unit circle.
Wednesday and Friday...Review of Laurent Series and Residues
Saturday AM OK, so the review is over. We will
return to the applications of residues to the computation of integrals on
Monday
25) page 325, 326 # 2 ( Joe),
6 ( Jenna), and 3( Leah) Tuesday AM I guess my only comment is that number three
involves a good deal of bookkeeping. The four zeros of x^4+1
all come into play. I computed them and labeled them z1,z2,z3,z4 by quadrant . The residue at z1 is z1^2 +1 divided
by z1-z2 times z1-z3 times z1-z4... whew!
26) page 392 #3 find the
two parts that are like what we did in class and do those.
27) Finish problem 3 and then finish the problem that we started about
solving the Dirichlet problem in a lens shaped
region.
Sunday AM
If 3e is giving you fits, you can try this: w = 1/z which equals x - iy divided by x^2 + y^2. From the
equation of our circle, x^2 + y^2 = 4x-3. So if you write w = u + iv, then you can solve for u and v in terms of x and y. Then
solve THOSE equations for x and y in terms of u and v (!!!) Then write out (x-2)^2 +y^2 =1 and see what equation you get for u and v. ( It’s
a circle!)
Don’t worry if you don’t get it done.