Math 621 Homework Assignment 4 Spring 2006
Due: Monday, March 27
1. Use Green’s Theorem to prove a weaker version of Cauchy-Goursat’s Theorem for
a rectangle:
Let fbe a holomorphic function defined and having a continuous derivative f0in
an open set Ucontaining a rectangle R. Then
Z∂R
fdz = 0.
Recall the statement of Green’s Theorem: Let γbe an oriented piecewise smooth
simple path (i.e., each connected component of γdoes not intersect itself) in the
plane. Assume that γbounds a region D(and has the induced orientation, i.e.,
each smooth piece of γis oriented so that Dis on the left as you move along
γ). Let p(x, y), q(x, y ) be two functions which are defined and have continuous
partial derivatives in an open set U⊂R2containing Dand γ. Then
Zγ
pdx +qdy =Z ZD∂q
∂x −∂p
∂y dxdy.
2. (a) Let Dbe an open disk in Cand let fbe continuous in D. Suppose that
Z∂R
f(z)dz = 0 for every closed rectangle Rcontained in D. Prove that fis
holomorphic.
(b) Suppose that fis continuous in all of Cand holomorphic in C\R. Prove
that fis holomorphic everywhere.
3. Let Ube an open subset of Cand fna sequence of holomorphic functions which
converges, uniformly on compact subsets of U, to a function f. Prove that fis
holomorphic in Uand that f0
nconverges, uniformly on compact subsets of U, to
f0.
4. Lang page 132 Problem 1: Find the integrals over the unit circle C:
(a) ZC
cos(z)
zdz (b) ZC
sin(z)
zdz (c) ZC
cos(z2)
zdz
5. Ahlfors page 120 Problem 3: Compute Z|z|=ρ
|dz|
|z−a|2under the condition |a| 6=ρ.
Hint: Make use of the equations z¯z=ρ2and |dz|=−iρ dz
z.
6. Show that the successive derivatives of an analytic function at a point can never
satisfy
f(n)(z)
> n!nnin two ways: (a) Using Cauchy’s Estimate. (b) Using
Taylor’s Theorem.
7. Lang page 132 Problem 3 (modified): Let fbe an entire function, ka positive
integer, and let kfkRbe the maximum of |f|on the circle of radius Rcentered at
the origin. Then fis a polynomial of degree ≤kif and only if there exist constants
Cand R0≥0 such that
kfkR≤CRk,
for all R≥R0. (Note: one direction was proven in HW 1 Problem 8).
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