MATH 440/508 - Assignment #2
Due on Wednesday October 6, 2010
Name —————————————–
Student number —————————
1
Problem 1:
Suppose f (z) and h(z) are both analytic on {|z| ≤ R} and f (z) 6= 0
on {|z| = R}. Prove that for some small > 0, functions f and f + h
have the same number of zeros in {|z| ≤ R} counting multiplicities.
2
Problem 2:
Let p(z) = z 6 + 9z 4 + z 3 + 2z + 4. (a) Determine the number of zeros
inside and outside the unit disk in each quadrant. (b) Show that the
zeros of p(z) that lie outside the unit circle satisfy |z ± 3i| < 1/10.
3
Problem 3:
For a fixed complex number λ, show that if m and n are large integers,
then the equation ez = z + λ has exactly m + n solutions in the
horizontal strip −2πim < Im z < 2πin.
4
Problem 4:
Suppose that f is entire and f (z) is real if and only if z is real. Use
the Argument Principle to show that f can have at most one zero.
5
Problem 5:
Sketch the closed path γ(t) = eit sin(2t), 0 ≤ t ≤ 2π , and determine
the winding number W (γ, ζ) for each point ζ not on the path.