Math 534 HOMEWORK 6 (due Wednesday, Nov 7) Autumn 2007
WARNING: some of the problems may be on material we studied earlier.
1(2 pts) Do problem 1(b,d,f) from Gamelin V.4 (p. 147).
2(2 pts) Do problem 4 from Gamelin V.4 (p. 148).
3(3 pts) Let fbe a power series which converges in the whole C. Prove that if
ZZC
|f(x+iy)|dx dy < ∞,
then f≡0.
4(3 pts) Suppose P∞
n=0 |an|2<∞. Show f(z) = P∞
n=0 anznis analytic in {z:|z|<1}.
Compute
lim
r%1Z2π
0
|f(reiθ |2dθ
2π.
(Prove your answer.)
5(3 pts) Do problem 14 from Gamelin V.4 (p. 149).
6(1 pt) Do problem 1(b,c) from Gamelin V.5 (p. 151).
7(2 pts) Calculate the terms through order 6 of the power series expansion of Log(sin z
z)
about z= 0.
8(4 pts total) (a) Do problem 6 from Gamelin V.6 (p. 154).
(b) Let Fn(z) = P∞
k=0 an,kzkand g(z) = P∞
k=0 bkzkbe entire functions. Suppose that |an,k| ≤
|bk|for all nand k, and that
lim
n→∞ an,k =ak
exists (for all k). Show that f(z) = P∞
k=0 akzkis an entire function and that fn→fnormally
in C. Show by means of an example that the assumption |an,k| ≤ |bk|cannot be dropped.
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