Math 448 Homework 10 Due Fri., Nov. 9, 2007
(ungraded) §3.1 – 1, 13
1. (graded) §3.1 –2. It is easier to do this by Rouch´e’s theorem. Let the “main” function
be the first and last terms, and the “perturbation” be the middle.
2. (graded) §3.1 – 20.
3. (graded) (E) Suppose fis entire and |f(z)| ≤ 1 on the unit circle |z|= 1. Using
Rouch´e’s Theorem, prove that there is exactly one z0with |z0|<1 and
f(z0) = 10z0
2 + z0
.
Hint: It will help to define φ(z) = 10z
z+2 .
4. (graded) (E) Let f(z) = 1
z+1
z−1. Find, carefully, Laurent series for fwhich converge
in each of the following regions: (a) |z−2|<1, (b) 1 <|z−2|<2, (c) 2 <|z−2|.
5. (graded) (E) Let Rdenote the (closed, solid) square in the complex plane with vertices
1±iand −1±i.
a. Determine the maximum and minimum of |ez|on R.
b. Determine the maximum and minimum of |z2|on R.
c. Use Rouch´e’s Theorem to determine the number of zeros of f(z) = ez+ 100z2in R.
6. (graded) (E) Determine the linear fractional transformation T(z) = az+b
cz+dso that T(0) =
1, T(1) = iand T(∞) = −1, and determine T(i).
7. (graded) What is the name of the author of the textbook. This is not a trick question,
but a way of thinning the homework.
8. (bonus) Suppose fis an entire function, f(0) = f′(0) = f′′ (0) = 0 and f′′′(0) = 1. Show
that there exists z0with |z0|= 2 and so that |f(z0)| ≥ 4
3. Hint: consider g(z) = z−3f(z)
(for z6= 0), and examine the singularities of gat 0.
9. (bonus) Find positive integers 0 < r1< r2with the property that the polynomial
f(z) = z5+ 2z3+ 4z+ 100 has no zeros in the region |z| ≤ r1and five zeros in the region
|z|< r2. Needless to say, this problem has more than one correct answer.
10. (bonus) Work the integral of §2.6 27 thorugh 31 with f(z) = 1
(3z−1)2and simplify your
final answer to get information about the sum of a subset of {1
n2}.