Homework assignments, Complex Variables
Janeba, Fall 2009
Jump to this week's assignments.
Week #1
- Sept. 2: Read Churchill sections 1,2 and do p. 5
#1,2,4,5,8a,10,11
- Sept. 4: Read sections 3-4, do p.8 #1,2,4,6,7
Week #2
- Sept. 9: Read sections 5-7, do p.12 #1ab,2,4,5 and p. 14 #1ab,2,
"3b", 5,12 (use #11 as suggested, without actually doing #11).
- Sept. 11: Do p. 15 #6,7,10a,14,15 and p. 23 #6,"9a".
You may want to read section 8 for the last part.
Week #3
- Sept. 14: Do p.23 #1,2,5,7
- Sept. 16:
- Turn in: p.5#4, p.8#1c,6, p.12#5, p.14-15#3b,5,7
- p. 29-30 #2,3,5,7,8 p.33 #1-3
- Prove, using our text's definitions:
- A set S in the complex plane is open iff all elements
of S are interior points of S.
- A set S in the complex plane is open iff the
complement of S is closed.
- Sept. 18: p.33 #7,9,10
Week #4
- Sept. 21: Read sections 13,14. Do p.37 #1-3, p.44 #1-5
- Sept. 23: Do p.44 #6,7 and p.55 #1,2a
- Sept. 25: Prove:
- The limit as z
-2i
of 5z+1 is 1-10i.
- The limit as z
3 of z2
is 9.
- If the limit as z
zo
of f(z) is w1 and the limit as z
zo of g(z) is w2,
then the limit as z
zo
of (f(z)+g(z)) is (w1+w2).
Week #5
- Sept. 28:
- Turn in: p.22#1b,2b,7,9a, p.29#2b,7, "Prove that a set S
in the complex plane is open iff SC is
closed," p.37#3, and p.44#6
- Do p.55#4,5,7,9, and write up a (more detailed) proof of
theorem (1) (just part (1)) on p.51
- Sept. 30: p.55#10,11,13 and show why, if
z
is defined as z - zo, that the limit as z
zo of f(z) is equal to the
limit as
z
0 of f(zo+
z). Try to be rigorous.
- Oct. 2: Quiz #1; Do: Find f '(zo)
from the definition of derivative for f(z)=3z2+2z,
and do p.62 # 1abc,2,3,5,7,9. Pay particular attention to the
methods called for -
when you get to use the
differentiation rules in section 20, cite them specifically one at a
time as you use them.
Week #6
- Oct. 5: Catch up on old homework and work on take-home
portion of exam 1, distributed today. Also, read section 21,22.
- Oct. 7: Do p. 71 #1a,c,2a,b,3a,b,5,6
- Oct. 9: Read section 24-26, do p.77 #1acd,2,3,4,7
Week #7
- Oct. 12: p. 92 #1-4,8ab,10,14
- Oct. 14: Exam #1
- Oct. 16: Read sections 29-32, do p.97 #1-4 (note the difference
between log and Log in these problems),6,8, p.100 #1
Week #8
- Oct. 19: p. 100 #2-4, read section 33, do p. 104 #1,2.
Also, draw a sketch of the (infinitely many) values of 2i.
Of course, you'll only actually plot finitely many; identify the
"trends."
- Oct. 21: p. 108 #1b,2,3,4a,7,11,12b
- Oct. 23: Mid-semester day (no classes)
Week #9
- Oct. 26: Read sections 37,38, do p. 121 #1a,2,3,5
- Oct. 28: Read section 39, do p. 125 #1,2,3 and prove that
if zo is a complex constant and w(t)
is a complex-valued function of t, then
zow(t)dt = zo
w(t)dt provided the latter integral
exists (hint: break into real and imaginary parts).
- Oct. 30: Read 40,41 (we won't worry about 42 just yet) do p. 135
#2,4,5,8
Week
#10
- Nov. 2: Read section 43, do p. 140 #1-3
- Nov. 4: Read section 44,45, do p.149 #1-3 (get the handout
from today if you haven't already)
- Nov. 6: prove the lemma, used in class today: If "-C"
is the contour given by traversing the contour C only
backwards, show that
-cf(z)dz [the integral of f
over -C] = -
cf(z)dz [the opposite of the
integral of f over C].
Hint: Suppose z(t), t
[a,b]
parametrizes C. Give an explicit parametrization of -C
in terms of z(t), and use it to calculate the first
integral.
Week #11
- Nov. 9: Evaluate the following integrals using your choice of:
parametrization, the antiderivative theorem (if applicable), or the
Cauchy-Goursat theorem (if applicable).
- f(z) = ez and C is
the contour from 0 to 2+
i
made of two segments: the first from 0 to 2, the second from
there to 2+
i.
- Same f and C as in (1), except this time C
is a single straight line segment from 0 to 2+
i.
- f(z) =3/(z-i) and C is the
circle |z-i| = 4, traversed counter-clockwise.
- f(z) =3/(z-i) and C is the
circle |z-10i| = 4, traversed counter-clockwise.
- f(z) =2z+1 and C is any contour
from -1 to 2i. (Include your explanation of why the choice of
contour doesn't matter.)
Solutions (check your work
before looking, you'll want the practice)
- Nov. 11: Quiz!
- Nov. 13: Work on your take home, due Friday 11/20. Extra
copies are pinned on the board next to my office door.
Week #12
- Nov. 16: Add these three problems to your take-home (due Friday): P. 171#4, p.179#2 (Wed.'s lecture will help here), and:
- Let C be the circle |z-i|=3, positively oriented, let f(z)=ez/(z3-4z2), find the integral of f over C.
- Nov. 18:
- Nov. 20: Take-home due Homework for Monday:
- Give a series for 1/(z-4) in powers of z (equivalently, "expanded about zero") that is valid inside some disk.
- Give another series for 1/(z-4) in powers of z-2 (equivalently, "expanded about 2").
- Do p. 188 #2,6,8 (for the latter two, try to follow some of the text examples, and ask on Monday).
Week #14
- Nov. 30: Read sections 60 & 62 (61 is for later), especially 62
examples 3-5 and the discussion that precedes them. Then do p.
205 #1,2 (hint: find the series for ez in powers of z-1 first.), 4,5,7
- Dec. 2: Do p. 219 #1,2,3,6 and p. 225 #1
- Dec. 4:
- Read sections 68-70.
- Do p. 239 1abce (for e,
first find the MacLarin series for sinh(z) remembering that the
derivative of sinh z is cosh z and the derivative of cosh z is sinh z
(no negative sign)).
- Do also p. 239 #2
- Check out the bonus problem I emailed all of you.
Week #15
- Dec. 7: Do p. 248 #1,4,5 and p. 318 #2,5 (turn in p. 248#4 on Wednesday)
- Dec. 9: Turn in p.248 #4
- Dec. 11: (Last day of class)
Week #16
- Thursday, Dec. 17th from 2:00 - 5:00 p.m: Final Exam.
Items of
general interest:
Free
advice on preparing for careers: this is supposedly aimed at
freshmen, but all students should find it useful
...from Keith Devlin, editor of the MAA Focus.
Also interesting:
Here's some stuff about Pythagoras.
The link is slow (it's from England), but it's worth the wait.
Stuff about Pythagoras
and music (did you know he invented the harmonic scale?) Really
cool
if you have a computer with sound capability.
Prof.
Janeba's Home Page | Send comments or questions to: mjaneba
willamette.edu
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of Mathematics | Willamette
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