Last handouts: Homework 11 Solutions
M 11/26 --
(i) We didn't decide on a time for the next test. It will either be
next Monday night, next Tuesday night or Monday in class. Obviously,
it would be a bit shorter if it were in class.
(ii) I went over problem 3.2--4 from the homework due today.
(iii) I proved the reflection principle (see p. 291), though with a
different proof.
(iv) I talked about conformality of mappings, the basic point being,
that if f is analytic and f'(z_0) = 0, and we consider two curves in
the complex plane, C_1 and C_2, which intersect at z_0 with an angle
of theta, and then consider the images f(C_1) and f(C_2), these will
intersect at f(z_0), also at an angle of theta. (See section 3.4.)
In particular, the images of the line x = x0 and y = y0 will be
perpendicula, and the images of the level sets Re(f(z)) = u0 and
Im(f(z)) = v0 will also be particular.
(v) Coming up on Wednesday: decision on the test, the return of HW11
and comments thereon, plus a summary of some topics from the syllabus
which won't be tested.
Homework 11 hints
A student writes: I am completely at a loss for 8, 9, 10.
My reply: Brief hints. 8. Let g(z) = (2+i)z^2 9. Think Schwartz lemma and factor
f. 10. T(T(z)) is also a fractional linear transformation.
This always happens this time of year; I slip up on postings, except for the homeworks itself. Here are two blocks of scanned sets of notes: Homework Solutions 7,8,9,10 and Extra HW comments and extra notes. This should catch us up through Break. If it isn't clear, the second test will cover everything since the first test, so that's sections 2.4 through 3.3, except 3.1.1. Ask questions in class if you aren't sure if a topic will be "fair game".
Out of order: Homework Eleven, due M 11/26.
Out of order: Homework Ten, due F 11/9.
Out of order: Homework Nine, due F 11/2.
General remark. This is a nasty assignment. I'll try to make up for it in HW9.
A student writes: I was looking at example 5 in the book. So they take this real function
and plug in z and make it an imaginary function.. but I don't understand
where they got the e^iz. It just looks like they replaced sinx with e^iz,
and I want to know how/why they did that, or where they get that from.
My reply: if you have a *real* function f(x) for real x and you integrate f(x)*e^{ix}
on the real axis, you are actually integrating the complex function f(x)*cos(x) + i
f(x)*sin x. This should serve as a hint for 2.6.4 as well.
A student writes: On (2.6 10): I obtained a quartic in the
denominator and I have no clue how to factorize it. Is there a trick/hint?
My reply: Hint: could it possibly be the square of a quadratic?
Out of order: Homework Eight, due F 10/26.
Out of order: Homework Seven, due F 10/19.
Test 1 will be W 10/10 in Room 141 Altgeld Hall at 7PM, covering material through section 2.3.
W 10/10 -- Here are the Extra Comments on hw 6. Otherwise, we reviewed for the test and continued our discussion of isolated singularities.
M 10/8 -- Few questions. Brief discussion of HW5 2.3 #14,15,16. Section 2.4 wrapped up. Here is the Homework Block, consisting of handwritten solutions to HW 3,4,5,6 and additional comments on HW 3,4,5. I'll try not to fall so far behind again. On HW 6, the book's solution for 2.4 #15 has a typo; the sum should begin at 0, not 1.
Catchup: F 9/28, M 10/1, W 10/3, F 10/5. Until the exam, all new material presented will be secondary to answering questions on the upcoming test. The only results on this material which are not in Fisher are the generalization of Liouville's Theorem on entire functions to |f(z)| < c * |z|^m for |z| > R implying that f is a polynomial of degree at most m (see 2.4, #21) and the theorem that if f is entire and |f(z) - w| > r > 0 for some w and r, then f is constant. I will scan all hw solutions and additional comments and link them on Monday morning.
Out of order: Homework Six, due M 10/8.
W 9/26. Homework 5 is now due on M 9/31, er 10/1. HW6 should still be out on Friday and due on F 10/5. We covered Cauchy's Theorem, and did a couple of specific examples. In passing, we now know an antiderivative for (1+x^4)^(-1).
M 9/24 -- We stopped at the very beginning of Cauchy's Theorem in 2.3. We have found the antiderivative of an analytic function in a simply connected domain. Homework 4 was returned with additional comments. The homework out on Friday (#6, due Oct. 5) will be the last one on which the first exam is based. No date for the test has been set.
W 9/19, F 9/21 -- We reviewed Green's Theorem and then started section 2.3 and Cauchy's theorem. The integral of an analytic function on a simple closed contour wants to be zero, if the function is also analytic on the interior of the contour. I will scan the homework four solutions when I get a chance, or ask me for them if you don't have them. Here is Homework Five, due F 9/28:
Homework 4 questions
A student asks:
I'm a bit confused on extra problem 8. So what am I supposed
to be seeing when I take z^2 * f(z)?
My reply: Think of z = x + iy.
A student asks: Does "closed form of the power series" mean the series
all tidied up in one expression?
My reply: Yup. The closed form for (1-2z)^(-1) + (1-z/3)^{-1} would
be the sum as n
goes from 0 to infinity of (2^n + (1/3)^n) * z^n. I guess I'm a bit sloppy
in my language -- I mean that you should find a specific expression for each
term of the power series. (That's what the last part of the question is
about.)
M 9/17 -- Homework 3 returned with comments and additional remarks, link to be put in later More on mappings; much more on series. A non-book example, elaborating on HW1 #10. We left in the middle of the proof that convergent power series are analytic. The proof will be completed Wednesday, along with a review of Green's Theorem.
F 9/14 -- We finished 2.1 (alternate proofs of an extended Theorem 2, p.82)) and began 2.2, on infinite series. We skipped section 2.1.1 on flow, although that might be very interesting to physicists and engineers.] Homework three collected and solutions, link to be put in later were distributed, as was Homework Four, due Friday, September 21.
A student writes on HW 3: I'm having a hard time seeing what you mean in
#14.
My reply: What I have in mind is this. Prove this first when P(z) is a
product of
one linear factor, then when it's a product of two linear factors. Then
use induction, assuming it's true for polynomials which are a product of n
linear factors, to show it's true for products of n+1 linear factors, so
the "n" refers to the degree of P.
W 9/12 -- More on differentiation, the Cauchy-Riemann equations and harmonic functions. A discussion of mathematical etymology, taken from Earliest Known Uses of Some of the Words of Mathematics.
M 9/10 -- The phone and email list (unlinked) was distributed. More discussion of HW2, with additional remarks distributed. We did a few examples of complex integration, including the path-independence of z^m and the path-dependence for 1/z, at least on two contours on the unit circle, from 1 to -1. And, we started complex differentiation.
F 9/7 -- Some discussion of mapping and a serious introduction to complex integration. Homework two collected and solutions were distributed, as was Homework Three, due Friday, September 14. Monday will have more complex integration, and a historical etymological explanation of modulus and absolute value.
Small correction to HW2: Problem 5(c) should read Log(z) = - 1 - Pi*I, not Log(z) = -1 + Pi*I; the only difference is in the change of sign.
W 9/5.-- HW 1 graded and returned with additional remarks. Finished discussion of 1.5, started 1.6. Friday will have HW2 due, HW3 out, and I'll distribute the email/phone voluntary sign-up list.
F 8/31 --. Distributed Homework One Solutions and Homework Two, due Friday, September 7. New material covered was most of section 1.5. We'll finish that on Wednesday and move on to section 1.6. Have a good break.
Questions on Homework 1
A student writes: In one of the problems we solve (z+1)^4 = 1-i . (I don't
know what cos(-pi/16) is!), are you okay with writing something like r
cos(theta) - beta + i(r sin(theta)) ? This also kind of relates to the
problem about showing the circle |z-r| = r under f(z) = 1/z gets mapped to a
vertical line. I also can't express that easily in polar form (which makes
taking 1/z really easy!), but I guess there is not a good way to fudge it
into polar, right? I'll just find z = a + ib, with a and b in terms of r and
use the more complicated formula... drat.
My reply: First, cos(pi/16) is a perfectly well-defined number! If you
want it in numerical
form, you can fiddle with the half-angle formulas from trig. Second, it is not
necessary to write every complex number in both ordinary and polar form.
W 8/29 -- A day of review of basic analysis notions: limits, continuity, sequences series. The key examples were T(z) = (1+z)/(1-z) and the geometric series.
M 8.27 -- A day of topology. Most of what was covered was in section 1.3. I also talked about when the {union, intersection} of two {open, closed, convex, starshaped, connected} sets must maintain that property and about stereographic projection, which leads to a "compactification" of the complex plane. You can find the definition of "star-shaped" in 1.3, HW 18; I'll eventually write something up about "stereographic". On Wednesday, I'll answer questions on the homework and cover 1.4.
F 8/24 -- We have now finished the first two sections of Fisher. Two handouts: How to Solve It guide and Homework One, due Friday, Aug. 31. On Monday, we will cover section three, on topological considerations.
W 8/22 -- First day of class. The basic arithmetic of complex numbers in three ``modes'' as a + i b where a and b are real and i*i = -1, as ordered pairs (a,b), and as points in the plane. Why i is not "the square root of -1". Handout: Course Organization. Informal homework: let a denote the square root of three and consider (a + i)^n for values of n up to about 20. Read sections 1.1 and 1.2 of the text and bring questions.