homework.html

COMPLEX ANALYSIS HOMEWORK
Dr. Rachel Hall
Spring 2006

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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 Thursday, January 26th

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 Thursday, February 2nd

Function diary. Your function went on vacation this week. Check your mail for postcards!

Problems from the book.

p. 21: 1, 2, 4, 6, 10 (first identity only, not Lagrange’s identity)
p. 28: 1, 2b, 3b, 6, 7
p. 31: 1

Other problems.

Finish the proofs of the BINGO! board identities (columns 1 and 3).
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 when n is odd.

Assignment #3, due Thursday, February 9th

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. 31: 4b, 5
p. 35: 1, 3, 4
p. 42: 1, 3, 4, 7

Assignment #4, part 1 due Monday, February 13th; part 2 due Thursday, February 16th

PART I: Sample test.

PART II:

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.

p. 53: 1ab, 3b, 5, 10, 11

Assignment #5, due Thursday, February 23rd

Test rewrites due.

Function diary.

Don’t forget to resubmit anything from the function diary that you had comments on before.
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, 4b, 5

Assignment #6, due Thursday, March 2nd

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?

Worksheets: Changes in F, The return of F, The revenge of F

Problems from the book.

p. 73: 1ad, 2abc, 4c, 7ab
p. 89: 1ab, 3, 6, 8ac, 10, 11

Assignment #7, due Thursday, March 16th

Function diary.

Your function went on Spring Break! Don’t forget to resubmit anything from the function diary that you had comments on before.

Problems from the book.

p. 94: 1, 2, 4, 5, 7
p. 96: 1, 2
p. 99: 1, 2bc, 3, 9

Assignment #8, due Thursday, March 23rd

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 (DOUG), 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 DOUG.

Other problems.

If your name is NOT Doug, 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. 100: 8ac
p. 104: 5, 13, 17, 18
p. 107: 2, 8, 7b, 13ab, 15b
p. 110: 2ab

Assignment #9, due Thursday, April 6th

Function diary.

· Find the contour integral of your function along the positively-oriented upper semicircle of radius 2 (that is, the contour 2e^it, 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 #10, due Thursday, April 13th

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.

Problems from the book.

p. 153: 1aef, 2ac, 3, 6, 7
p. 162: 1, 2, 3, 5

Assignment #11, due Thursday, April 20th

Know thy analytic functions.

Complete questions 1-19 in the 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. 171: 1, 4, 5

Assignment #12, due Thursday, April 27th

Function Diary (due May 5th). A complete list of problems is now posted. Have at it!

Problems from the book.

p. 181: 3, 6
p. 188: 1, 2, 5, 10, 13
p. 198: 1, 2, 3, 4, 5