Math 246A Spring 09
Homework 2
Due Thursday, May 7 in Section
In these problems let Ddenote the open unit disc and Dthe closed unit disc.
For a function fwrite Mf(ρ) = sup{|f(z)|;|z|=ρ}.
1. Show that the radius of convergence rof the power series P∞
n=0 anzncan
be computed from
r−1= lim sup |an|1/n.
2. Let fbe function that is analytic on a domain Dand that is not
identically zero. Show that the set of zeros of fin Dis at most
countable.
3. Define the coefficients anby
f(z) =
∞
X
n=0
anzn= (1 −z−z2)−1.
Show that a0=a1= 1 and an=an−1+an−2for n≥2.
4. Suppose that fis analytic in an open set containing Dexcept for a pole
at z0on the unit circle. Show that if
f(z) =
∞
X
n=0
anzn
in Dthen
lim
n→∞
an
an+1
=z0.
Observe that in this case the radius of convergence is limn→∞ |an
an+1 |but
we have more information by not taking absolute values.
5. Let
f(z) =
∞
X
n=0
anzn
have radius of convergence r > 0. Show that for each ρwith 0 < ρ < r
we have that ∞
X
n=0
|an|2ρ2n≤Mf(ρ)2.
Observe that this implies the bound
|an| ≤ Mf(ρ)ρ−n,
which follows from the Cauchy integral formula.
(Hint: Integrate z−1f(z)f(z) over a circle.)
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