Download Math 611 Assignment # 4 1. Suppose C is a boundary of a simply

Math 611
Assignment # 4
1. Suppose C is a boundary of a simply-connected domain D ⊂ C oriented in the counterclockwise direction. Recall the argument principle: If f (z) is meromorphic inside D
and is analytic and not equal to zero at all points on the boundary C. Then
I 0
1
1
f (z)
∆C arg(f ) =
dz = N0 − Np ,
2π
2πi C f (z)
where N0 is the number of zeros (w.r.t. algebraic multiplicity) inside D and Np is the
number of poles (w.r.t. pole orders) inside D.
Now, consider ξ ∈ C and a function f (z) meromorphic inside D and analytic on the
boundary C. Suppose f (z) 6= ξ and f (z) 6= 0 for all points z on the boundary C.
Suppose also f (z) 6= 0 for all z ∈ D (i.e. N0 = 0). Explain the meaning of
I
I 0
f 0 (z)
f (z)
1
1
dz −
dz
2πi C f (z) − ξ
2πi C f (z)
2. How many roots of ez − 3z 3 are there inside the unit circle B1 (0) around the origin?
Hint: Find f and g such that f (z) + g(z) = ez − 3z 3 and |f (z)| > |g(z)| for all z in
{z : |z| = 1}. Use Rouché’s Theorem.
3. How many zeros does the polynomial z 7 + 4z 4 + z 3 + 1 have in the regions {|z| < 1}
and {1 < |z| < 2}? Hint: Compare |z 7 + 4z 4 | to |z 3 + 1| on {|z| = 1}, and use Rouché’s
Theorem.
4. Show that all five roots of the algebraic equation z 5 + 15z + 1 must be situated in the
interior of the circle |z| < 2, but that only one root of this equation is in the circle
|z| < 32 .
5. Show that the equation
zea−z = 1,
where a ∈ R and a > 1,
has precisely one root in the circle |z| ≤ 1.
6. Show that, in |z| < 1, the function
f (z) = z +
takes every nonreal value exactly once.
1
z
7. Let f (z) be analytic in a domain D and on its boundary C. If |f (z)| = 1 for all z ∈ C,
show that unless f (z) is a constant there must be at least one zero of f (z) in D.
1
Hint: Consider f (z)
.
8. (1998 qual, # 2) Let f (z) be analytic function in the entire complex plane. Suppose
there exist real numbers M and α ≥ 0 such that |f (z)| ≤ M (1 + |z|)α . Prove that f (z)
is a polynomial of degree less than or equal to α.
9. (1998 qual, # 3) Let f (z) and g(z) be analytic functions in the open disc |z| < 2.
Assume (a) |f (z)| ≥ |g(z)| for any z with |z| = 1 and (b) f (z) is not zero for any z
with |z| < 1. Prove that |f (z)| ≥ |g(z)| for any z with |z| < 1. Give an example which
shows that this conclusion is not true without assumption (b).
10. (2005 qual, # 3)
(a) Suppose a function f is analytic everywhere in the complex plane except for a
finite number of singular points interior to a circle Γ, |z| = r, r > 0, which is
traversed once in the counterclockwise direction. Suppose Res{f (z), z0 } denotes
the residue of f (z) at z0 . Prove the equality
I
1
1
f (z)dz = 2πi Res
f
, 0 .
2
z
z
Γ
(b) Suppose P (z) and Q(z) are two complex polynomials of degrees n and m respectively. Suppose Γ is a simple closed contour that encloses all zeros of Q(z). If
m ≥ n + 2 show that the contour integral
I
P (z)
dz = 0 .
Q(z)
Hint: do a change of variable w = z1 .