COMPLEX ANALYSIS HOMEWORK
Dr. Rachel Hall
Spring 2010
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General Homework Policies. You
should start working on the homework problems for a section as soon as that
section is covered in class. Although you are encouraged to consult with
other students and seek help from me, your homework should ultimately represent
your own work. For more information, see the
department’s Academic Honesty guidelines. Answers unsupported by work will not receive credit.
Homework should be neatly handwritten or typed, on one side of the page
only. Please remove messy edges
and staple.
Assignment
#1, due Wednesday, January 27th
Function Diary. Your first assignment is to describe your function as
a function of real numbers. That is, investigate y = f(x) when x is
real. Your report should include
- Identification
of the domain and range and asymptotic behavior (What happens when x goes to plus or minus infinity? Are there
any vertical asymptotes?)
- Graph
of the function in the xy plane.
Problems from
the book.
- p.
4: 1-5, 10
- p.
7: 1, 2
- p.
11: 1, 4
- p.
13: 1, 2, 3, 6, 7, 10, 11
Assignment #2, due Wednesday, February 3rd
Function diary. Your function went on vacation
this week. Check your mail for
postcards!
Problems.
- p.
21: 1, 2, 4, 6, 10 (first identity only, not Lagrange’s identity)
- p.
28: 1, 2b, 3b, 6, 7
- p.
31: 1, 4b, 5
- After
you have done p. 28 #7, draw some typical pictures of the nth roots of unity on the unit circle if n is even. Give a geometrical explanation
for why the sum of the roots is zero when n is even. For extra credit, give a
geometrical explanation for why the sum of the nth roots is zero for any n, even or odd.
Assignment #3, due Friday, February 12th
Function diary.
- Find
the domain of definition of your function. Is it open? closed?
connected? We haven't discussed all the functions yet, so don't
hesitate to ask me if you aren't sure.
- Write
your function in either form (1) or form (2) on p. 34, as appropriate.
- Draw a
plot of the modular surface of your function. You can either use
Maple or do it by hand. If your function has any poles, find their
coordinates. For an explanation, go here (HTML) or here (Maple).
Problems from the book.
- p. 35: 1, 3, 4
- p. 42: 1, 3, 4, 7
Assignment #4, due
Friday, February 19th
Function diary.
- Find
the limit of your function at infinity. If the limit is undefined,
explain why.
- Are
there points at which lim_(z -> z0) f(z) =
infinity? To answer this question: (1) If your function
is defined on the entire complex plane, you need only check the limit at
infinity. (2) If your function is defined on a punctured
plane, find the limit(s) of f(z) at all the punctures.
Problems from the book.
Assignment #5, due
Friday, February 26th
Function diary.
- You
can resubmit anything from the function diary that you had comments on
before. At this point I’m not
grading them but I’ll let you know if there’s a problem.
- Where
does your function satisfy the Cauchy-Riemann equations? Some of you
will need to use the polar form (sec. 22).
- What
is the derivative of your function, and where is it defined?
Problems from the book.
- p.
59: 1, 3, 8a
- p.
68: 1cd, 2ab, 3abc, 4ab, 5, 7, 8
Assignment #6, due
Friday, March 5th
Function diary.
- Don’t
forget to resubmit anything from the function diary that you had comments
on before.
- Is
your function analytic on its domain of definition? Is your function
entire?
- Does
your function have any singularities? If so, where are they?
Problems from the book.
- p.
73: 1ad, 2abc, 4c, 7ab
- p.
89: 1ab, 3, 6, 8ac, 10, 11
Assignment #7, due Friday, March 19th
Function diary.
Find the largest domain in the complex plane on which your function is
analytic. Remember, this should be an open set. If you have a
multiply-defined function (TYLER), use the principal value of your
function. Prove that your
function is continuous on this domain (you should have already checked that the
function is analytic in Assignment 5, but if you haven’t, do so now). You
can use facts like “compositions of continuous functions are continuous” and
other results from Sections 15 and 17. If you think your function has any
discontinuities, explain why these are indeed discontinuities. This
assignment is pretty much a summary of things we’ve done before, unless your
name is TYLER.
Other problems.
- If
your name is NOT Tyler, prove that Log z is discontinuous at the negative real axis (you
might want to read back to the section on limits to see how you prove a
function is discontinuous).
Problems from the book.
- p.
94: 1, 2, 4, 5, 7
- p.
96: 1, 2
- p.
99: 1, 2bc, 3, 8ac,
9
- p.
103: 1, 2
- p.
107: 8
- p.
110: 2
Assignment #8, due
Monday, March 29th
Function diary.
· Find
the contour integral of your function along the positively-oriented upper
semicircle of radius 2 (that is, the contour 2eit, where t
goes from 0 to pi).
· Find
an antiderivative of your function.
Where is this antiderivative defined? continuous?
analytic? do you have to choose a branch of log to define an antiderivative?
· If
your function has a singularity, find a contour integral on a counterclockwise
closed contour around (one of) the singular point(s). Choose a crazy
contour that only contains one singular point.
· If
your function does not have a singularity, find (1) the contour integral
on a crazy counterclockwise closed contour around the origin and (2) the
integral on a contour from i to -i.
Problems from the book.
- p.
115: 2ab, 4
- p.
129: 1, 2, 3, 6, 10
- p.
141: 2, 5
Assignment #9, due
Wednesday, April 7th
Function diary.
- Your
function is tired and needs a rest.
Don’t forget to resubmit anything from the function diary that you
had comments on before. Here is a link to the complete Function
Diary assignment from 2006.
Problems from the book.
- Read
section 46. This section
states several results that follow from the Cauchy-Goursat Theorem. The proofs are relatively
straightforward (especially when compared to the C-G Theorem).
- p.
153: 1aef, 2ac, 3, 6, 7
- Extra
Credit Project on log and Log.
The idea is to make a list of identities about complex logarithms,
together with a page reference or short proof, plus “things that we expect
to be true but aren’t,” with counterexamples. Bring your list to class and whoever wants to earn
extra credit can figure out how to compile the lists. If you come up with something
good, I’ll type it for you!
Assignment #10, due
Wednesday, April 14th
Know thy analytic functions.
- Complete
questions 1-19 in this handout. Hint: pay attention to whether the domain referred to in a
theorem must be simply connected, or if any domain will do.
Function diary.
- Cook
up a Cauchy Integral Formula problem involving your function and solve
it. If your function can be
written as f(z) = g(z)/(z-z0)n , you can use the CIF directly; otherwise you’ll
have to be a little more creative.
Problems from the book.
- p.
162: 1, 2, 3, 5
- p.
171: 1, 4, 5
Assignment #11, due Wednesday, April 28th
Function Diary (due Friday, May 7th). A complete list of problems is now
posted. Have at it!
Problems from the book.
- p.
181: 3
- p.
188: 1, 2, 3, 5, 10
- p.
198: 1, 2, 3, 4, 5