NEWS UPDATES for 18.04: Complex Variables with Applications
 Spring 2000; Bazant and Rosales.

 THIS FILE WILL CONTAIN NEWS AND UPDATES REGARDING THE COURSE.
 >>>>>>>>>>> CHECK IT FREQUENTLY <<<<<<<<<<<

 Prof. Bazant's Office Hours: Wednesdays 3:30-5:00pm in Room 2-363B.
 Prof. Rosales' Office Hours: Tuesdays 3:45-4:45pm Room 2-337.

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 Tuesday, April 11, 2000:

 Please note (once again) that Exam #2 will be held this Friday, April 14  
 in Walker Gym from 12-1. It will cover all the material since the first
 exam up to yesterday's lecture on summation of series. Looking at your  
 syllabus, this means lectures 10-24 on contour integration, Laurent/Taylor
 series and residue calculus.

 Note that Drop Date is next Thursday April 20. We will have the exams
 graded by Wednesday April 19, so all your grades will be available at that
 time. I will be out of town next week but should be able to discuss your
 grades with you by email (and of course, you could also speak with Prof.
 Rosales).

 Tomorrow, I will begin lecturing on a new topic: Conformal Mapping and
 Harmonic Functions (which we have already discussed to some extent earlier
 in the course). There will be no lecture next Monday (Patriot's Day). Next
 Wed (4/19) & Fri (4/21) and the following Mon (4/24), you will be treated
 to three more lectures on conformal mapping by Prof. Toomre while I am out
 of town. The next wed (4/26) I will begin the final topic of Fourier   
 Series and Tranform Methods. That Friday the final problem (4/24) set is
 due.

 I will hand out PS#4 after the exam this Friday so you have two weeks to
 work on it. -MZB

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 Monday March 13, 2000:

 I'm sorry to say there is an error in the PS#2 solution for problem 4
 which I handed out in class today, as several students pointed out: Since
 v/2 actually goes from 1 to 2, C_w goes around the rectangle -1 < x < 1, 2
 < v <4, which has area 4, not 2. Therefore, the answer is 4i, not 2i. I
 was trying to cook up this problem to be extremely easy (when you saw how
 to do it!), so I apologize for inadvertently throwing in an annoying
 factor of two. "Minus one for the professor" once again! :)  -MZB

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 Wednesday March 6, 2000:

 The FINAL EXAM will be held from 9am-12noon in Walker Gym on Monday,
 May 15.

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 Wednesday Feb 23, 2000:

 1. Prof. Bazant's office hours have been moved to Wednesdays 
 3:30-5:00pm in 2-363B to avoid conflicting with recitations.

 2. Note that Math. Dept. Tutoring Room 2-102 is open from 3-5 and 
 7:30-9:30pm Monday through Thursday. No appointment is necessary.

 3. Exam times and homework due dates are not negotiable. If you miss 
 one, you will get a zero. If your excuse is very good and involves an
 unforeseen circumstance, Prof. Bazant will take it into consideration 
 when assigning your final grade.  Foreseen conflicts with extra-
 curricular activities do not provide valid excuses since schoolwork 
 is your first priority as a student. The exam and homework dates
 were clearly announced at the first lecture and on the syllabus.
 By registering to take this class for credit you are agreeing to meet 
 all its deadlines.

 _____________________________________________________________________
 Thursday Feb 10, 2000:

 HINTS on PS#1:

 Problem 1. Roots solve the equation P(z)=0, which can be written as
 (z-1)^6 = -(z+1)^6. This equation means that both the lengths and the
 angles of the complex numbers on either side are equal. My hint for part
 (a) is to think about the lengths, i.e. the moduli |z+1| and |z-1|.
 For part (b), try taking the ratio of the two sides and taking a
 root... (Make sure you get six solutions.)

 Problem 2. Think about the product a*z if a is positive and real. What
 does that mean? a*z is simply stretched by a factor a in length, with the
 same angle as z. Recall from lecture how the geometric series
 1+z+z^2+z^3+... looked like a "spiral chain" of vectors in the complex
 plane. Here, since a_k>0, it's very similar, only we don't know the length
 of each vector a_k z^k. However, we do know the angles... Draw a
 picture. Convince yourself that if |arg z| < pi/n, then the sprial has no
 chance to "close up" and reach the origin. (Recall that a "root" happens
 when the chain ends up at zero.)

 Problem 3: Recall what I did in class for geometric series, S(z) =
 1+z+z^2+...+z^n:  we multiplied S(z) by z, and got almost the same series
 back, so that when we subtracted S(z) - zS(z) we got something very
 simple, 1 - z^{n+1}. Then S(z) is determined as a ratio of two differences
 (not unlike the ratio of two *sines*) I'm asking for here. In this case,
 think of two complex numbers z_1 and z_2 such that z_1*S(x) - z_2*S(x) =
 (z_1 - z_2)*S(x) something simple, like a sine. If z_1-z_2 is also a sine,
 then you're in basically there... right?

 Problem 4: DON'T WRITE DOWN THE COMPONENTS OF EACH VECTOR (e.g. z=a+bi)!
 That's very messy and hard to complete. The whole thing can be done very
 quickly using algebra with complex numbers. Let each side of the
 quadrilateral (AB,BC,...) be a complex number, say a,b,c,d. What is
 a+b+c+d? You need to express the two segments of interest, EH and FH, in
 terms of a,b,c,d. At several points in the problem use the fact that i*z
 is a vector perpendicular to z with the same length as z. (As I stressed
 in class, multiplication by i is simply a rotation by 90 degrees.)

 _____________________________________________________________________
 Thursday Feb 10, 2000:

    Prof. Bazant will hold a special office hour at 4:30 today to provide
    help on the problem set. Yesterday's office hours were cancelled due
    to a building evacuation. 

    NOTE: Problem set #1 is due tomorrow at noon in 2-108.

    ****Late problem sets will not be accepted.****

 _____________________________________________________________________
 Wednesday February 9, 2000:

    Today's recitation was cancelled due to an accident that forced
    the evacuation of building 2 for 2 hrs.

    Feel free to come to any of the recitations tomorrow (Thursday)
    2:00-3:00 PM and 3:00-4:00 PM in rooms 2-131/132

    In fact feel free to come to as many recitations as you want. I
    do not mind if you switch them around, or come twice or whatever.
    Remember that a good chunk of the recitations changes from one
    session to the next, due to the different questions that may be
    asked in them.
 _____________________________________________________________________