Homework assignments, Complex Variables

Janeba, Fall 2009

Links of general interest.  

• 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
  

• 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.
  

• 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
  

• 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 z3 of z² is 9.
• If the limit as zz_o of f(z) is w₁ and the limit as zz_o of g(z) is w₂, then the limit as zz_o of (f(z)+g(z)) is (w₁+w₂).

• 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 S^C 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 - z_o, that the limit as zz_o of f(z) is equal to the limit as z 0 of f(z_o+ z).  Try to be rigorous.
• Oct. 2: Quiz #1; Do: Find f '(z_o) 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.

• 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
  

• 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
  

• 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)

• 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 z_o is a complex constant and w(t) is a complex-valued function of t, then z_ow(t)dt =  z_ow(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
  

• 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.

• Nov. 9: Evaluate the following integrals using your choice of: parametrization, the antiderivative theorem (if applicable), or the Cauchy-Goursat theorem (if applicable).
1. f(z) = e^z 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.
2. Same f and C as in (1), except this time C is a single straight line segment from 0 to 2+i.
3. f(z) =3/(z-i) and C is the circle |z-i| = 4, traversed counter-clockwise.
4. f(z) =3/(z-i) and C is the circle |z-10i| = 4, traversed counter-clockwise.
5. 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.

• 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)=e^z/(z³-4z²), find the integral of f over C.
  
• Nov. 18:
• Nov. 20: Take-home due  Homework for Monday:
  
1. Give a series for 1/(z-4) in powers of z (equivalently, "expanded about zero") that is valid inside some disk.
2. Give another series for 1/(z-4) in powers of z-2 (equivalently, "expanded about 2").
3. Do p. 188 #2,6,8 (for the latter two, try to follow some of the text examples, and ask on Monday).

• 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 e^z 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.
  

• 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)

• Thursday, Dec. 17th from 2:00 - 5:00 p.m: Final Exam.

Also interesting:

Stuff about Pythagoras and music (did you know he invented the harmonic scale?) Really cool if you have a computer with sound capability.

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