Homework Assignments, Complex Variables

Homework assignments, Complex Variables

Fall 2017

Dates shown are when the assignment is made, not when it is due.

Jump to current Week

Week #1

Assignment to prepare for first class meeting

Aug 28: Do Sec. 2 (p. 4-5) #1,3,4,5,11 and Sec 3 (p. 7-8) #1,2,5,7; read sections 4-6 and the class syllabus for discussion Wednesday.

Optional enrichment: read Wigner's THE UNREASONABLE EFFECTIVENSS OF MATHEMATICS IN THE NATURAL SCIENCES (1960).
Note, the math and physics both range very widely, from Galileo's law of falling bodies and the second derivative, to vectors in Hilbert space and quantum mechanics, but if you keep at it, you can get a sense of Wigner's argument. Jumping to the section THE ROLE OF MATHEMATICS IN PHYSICAL THEORIES may allow you to get the central point, or you can just read the summary on Wikipedia (along with the gist of some good formalist and Platonist critiques)

Aug 30:

Do Sec. 5 (p. 13-14) #1,3,5,6
and Sec. 6 (p. 16-17) #1,2,3b (i.e. property (4)), 5,10a (note that you must prove both directions), 15;
read sections 7,8 and prepare to discuss them Friday.
added later Wednesday: For review, print out this diagram and fill in the angles and coordinates of all the remaining marked points on the unit circle. Note that the unlabeled quadrants are symmetric copies of the labeled quadrant; use this to work out the angles and coordinated.

Sept 1:

Do Sec. 9 (p. 23-25)
#1, 2 (for 2b, give a symbolic argument to complement the geometric one given in class),
#5, 6, 9a (through the first identity, then show how if z = e^i(theta) then the real part of 1+z+z²+... gives the left side of Lagrange's identity.)
#10

At the start of class next Wednesday, turn in:

p.5 #5,11
p.8 #5,7
p.13 #3,5
p. 16 #3b,10a
p.23 #1, p.25 #10

Oh, yeah, and read sections 9-11 for next Wednesday

Week #2

Sept 4: Labor Day (no classes)

Sept 6:

Read sections 12, 13 for Friday
Do Sec. 11 (p. 30-32) #1,2,3,6 (BEWARE the typo in problem #6 some editions! It should read "... to factor z⁴+4 into quadratic ..."), 7, 8
Study question for section 12: Draw a few sets in the complex plane: some open, some closed, some neither, and label them. Can you draw a set which is both open and closed?

Sept. 8: Read sections 13,14,15 for Monday; Do section 12 (p. 34-35) #1-6,8

Week #3

Sept 11:

Read sections 15-17 for Wednesday

Do Supplementary Topology Problems #12-14,18 (problems available on WISE)

Do Sec. 14 (p. 43-44) #1-3,5,7,8 (Note the second answer provided in the text for problem #1 is really for part (c)!)

Sept 13:

Read sections 16, 17, 18 for Friday

Do Sec. 18 (p. 54-55) #1a,b 2a,c 5

Note the quiz on Friday is open-notes, but only hand-written notes are allowed; computer-printed or photocopied notes will be confiscated

Sept. 15: Quiz 1

Read sections 19-21 for Monday

Do Sec. 18 (p. 54-55) #3,7,9,10

Week #4

Sept 18: Read sections 19-21 again, and

Do Sec. 18 (p.54-55) #8,11
Start organizing for Friday's exam:

You may bring and use one normal sheet of paper with handwritten notes on one side, but no other resources.

Do sec. 20 (p. 61-62) #1,2ac,5,8

Sept 20: Review for exam:

make a list of 8 problems that have a good chance of showing up on the exam, and practice them.

Prep your page of notes. Is it reasonably complete?

Can you explain in written sentences what you are doing in the 8 problems? Have you practiced writing that explanation?

Further practice problems for differentiation: Use the limit definition of f '(z) to find f '(z) when:

f(z) = 1/z²

f(z) =3z+5

f(z) = 42i

f(z) =(2+i)z²+3z+5i

Can you explain the finer points of the derivations above, in coherent sentences?

Sept. 22: Exam I

Read sections 21 & 22 - possible open-notes reading quiz on Monday

Do Sec 24 (p.70-72) #1acd, 2b (for 2b, you will primarily use the big theorem in section 23, atop p. 66, which tells you what to do)

Week #5

Sept 25:

Read sections 23 & 24;

Do p. 61 #5,7; Do p. 70-71 #3, catch up on the p.70-71 homework from last Friday, and:
If f(x+iy) = e^ixe^-y + e^-ixe^y, at what values of z = x + iy is f differentiable?

Notes: Be careful computing u(x,y) and v(x,y) at the start - easy to lose minus signs. Remember that e^ix is not real, even though x is.
Be explicit with the Cauchy-Riemann fine print. You may use the fact that for real variables, sin(x), cos(x), and e^y are continuous at all values of x and y, and that the sum, product, and composition of continuous functions is continuous.

A review problem: For any one of the following, write up a polished presentation with all complete sentences; you are welcome to solicit feedback on your work during my office hours:

Prove that the limit as z goes to 4+2i of f(z)=2iz-8i is -4 using a formal - proof.
Find the derivative of f(z) = 3z²-5z+2 using a limit of a difference quotient (the limit definition of derivatives), showing all the steps neatly and carefully
Prove, using only the definitions of open set and interior point (and boundary point and exterior point, as needed) that a set S is open if and only if (prove in both directions) every point in S is an interior point of S.

Sept 27:

Read sections 25-26

Do Section 24 (p. 70-71) #6 (use the results of problem #5)

Do section 26 (p. 76-77) #1ab (use the fact that the derivatives of sinh(x) and cosh(x) are cosh(x) and sinh(x), respectively, for all values of x, and that these functions are all continuous everywhere.)

Do section 26 (p.76-77) #2,3,4a,6

For Friday, turn in: Section 26 #6,2a,1b, Sec. 24 #3b, any one of the "review problems" from Sept. 25, written up nicely (handwritten is OK), and section 18 #1b

Sept. 29: Oops, forgot to post homework. Sorry to disappoint.

Week #6

Oct 2:

Read sections 30-34 (30 & much of 31 are review of today's lecture)

Do section 30 (p.89-90) #1,2,3,6,8bc,9

Do section 33 (p. 95-97) #1,2b,3,

Oct 4: [Posted 1:55 pm 10/5] Read sections 35,36

Do section 33 (p. 95-97) #3c,6,8

Do section 34 (p. 99-100) #1,3

Oct. 6: Quiz 2

Read sections 41-43 (yes, we are skipping the trig functions for now)

Do section 36 (p.103) #1,2,8ac (note carefully the conditions on n)

Do section 42 (p.119-120) #1,2

Week #7

Oct 9: Read sections 44,45

Do section 42 (p. 119-120) #1,2 (if you haven't already), and #3

Do section 43 (p. 123-125) #1,5

Oct 11:

Do section 46 (p. 132-135) #1,2,3,5 (3 is long, just make sure you can set up the parameterized integrals; for 5 you can use a line segment, but try to see why the specific contour doesn't matter)

Oct. 13: Exam II

Week #8

Oct 16: Read sections 47-49, do section 46 (p.132-135) #6,9,13, Do section 49 (p. 147) #1,2,3

Oct 18: Read sections 50-51, do section 47 (p. 138-140) #1,2,3,4, Do section 49 (p.147) #5

Oct. 20: Mid-Semester Day (no classes)

Week #9

Oct 23: Read sections 52-54, do section 53 (p. 159-162) #1abcf, 5, 6, 8

Oct 25: Read sections 54-56, do section 53 (p. 159-162) #2,3,7(may contain Green's theorem), and...

Turn in on Friday, Oct. 27: p.133 #2, p. 138-9 #2,3 p.147 #2bc p. 159 #3, p.161 #6,

Oct. 27: (Today only, class moved to Eaton 110; the Board of Trustees will be using our regular classroom)

Read sections 57-59, do section 57 (p. 170-172) #1abce, 2, 3, 4, 5, 7

Week #10

Oct 30: Read sections 60-61, do section 57 (p. 172) #10, section 59 (p.177-178) #1,2,3

Nov. 1: Read sections 62-64, do section 61 (p. 185-6) #1,3,8 and...

Find a series involving z's whose sum is 1/(z+3), by manipulating the example "1/(1-z) = 1+z+z²+...+zⁿ+... whenever |z|<1". Be sure to state the set of values of z for which your series sums to 1/(z+3); there are multiple correct answers depending on the set of z values for which the result holds.

Nov. 3: Quiz 3

Exam 3 is take home, assigned today, due next Friday .

Week #11

Nov. 6:

Read sections 63-66

Work on take home exam assigned last Friday

Nov. 8: Read section 68, skim 69-72, and continue to work on Take Home

Nov. 10: (take home exam assigned last Friday due today at 3:00 p.m. at my office).

Week #12

Nov. 13:Read sections 74,75,76

Nov. 15: Read sections 78-80.

For Friday: Turn in the following problem at Friday's class:

Let C be the positively-oriented boundary of the rectangle -1 [< or =]

3, -4

4, and find [integral]

_Cf(z)dz , where f(z) = e^z/[(z²+4)(z-3)²] (that's the integral on the contour C of f(z)dz ) by finding the relevant residues.

Hint: The

business in section 80

Note: No need to simplify the complicated arithmentic in your answers

Note: Try to check your work on one or more residues by a clever use of the (possibly extended) Cauchy Integral Formula.

Nov. 17:

Thanksgiving Break (Nov. 20-24)

Week #13

Nov. 27: Read sections 85-87 (improper integrals via residues) and section 96.

Do section 86 #2,5 optional: #3; hint for #3: z⁴+1 = (z-₁) (z-₂) (z-₃) (z-₄) where ₁, ₂, ₃, ₄ are the distinct fourth roots of -1.
Do section 88 #1

Nov. 29: Read sections 97,98,99 in Churchill.

Do section 96 (p. 301) #1-4, look at section 98 (p. 305-306) #2,8,11

Dec. 1: Quiz 4

Read section 100 in Churchill

Do Section 98 (p. 304-306) #3,8,11 and section 100 (p.311-313) #5,6,10 and:

for a generic LFT f(z)=(az+b)/(cz+d) (remember the condition on a,b,c,d), at what values of z is f analytic? At what values of z is the derivative non-zero?
It turns out that LFT's have "square roots" in the sense that for any LFT f(z), there is another LFT g(z) such that g(g(z)) = f(z). (Note there is no actual "squaring" involved here.)
Start with g(z)=(1+z)/(1-z) and find g(g(z)), which shows that the "square root" of <what> is <what>?
Notice that g sends 0 1 -1 0. What rotation of S² could g be? What are the fixed points of g?

Week #14

Dec. 4:

Thinking of 1/z as a rotation of S² about the real axis by 180 degrees, if we wanted h(h(z)) = 1/z, we would want h to send 0 where?

where? Find two more points and where we want h to send them.
Then find h(z) in general.

Do section 100 (p.311-313) #1,3,7,9,12

Dec. 6:

Dec. 8: (Last Day of Class)

Week #15

Thurs, Dec. 14, 8-11 a.m. Final Exam

Friday, Dec. 15: Star Wars opens.

Please mail comments or questions to: mjaneba@willamette.edu

Willamette University Home Page