Math 534 Homework #1
Autumn 2008
Some are routine, but take time. A creative idea on others can lead to simple solutions.
Learning how to write accurate, succinct and subtantiated arguments is part of this course, so
please write your solutions carefully. You may discuss the problems with others, but you must
write the solutions in your own words.
1. Prove that fhas a power series expansion about z0with radius of convergence r > 0 if and only
if g(z) = f(z)−f(z0)
z−z0has a power series expansion about z0, with the same radius of convergence.
(How must you define g(z0), in terms of the coefficients of the series for fto make this a true
statement?)
2. For what values of zis P∞
n=0 nznconvergent? Same question for
∞
X
n=0z
1 + zn
.
Draw pictures for the regions.
3. Define ez=P∞
n=0
zn
n!. Show that this series converges for all z, that ezew=ez+w, that
eiθ = cos θ+isin θwhere cos θand sin θare defined by their series expansion as you learned them
in calculus. Finally show |ez|=eRez.
4. Suppose P∞
j=0 |aj|2<∞. Show f(z) = P∞
j=0 ajzjis analytic in {z:|z|<1}. Compute
lim
r%1Z2π
0
|f(reiθ )|2dθ
2π.
(Prove your answer).
5. Prove the parallelogram equality:
|z+w|2+|z−w|2= 2(|z|2+|w|2).
In geometric terms, the equality says that the sum of the squares of the lengths of the diagonals of
a parallelogram equals the sum of the squares of the lengths of the sides. It is perhaps a bit easier
than a proof using high school geometry.
6. Suppose that fis a continuous complex valued function on [a, b]. Let
A=1
b−aZb
a
f(x)dx,
be the average of fover the interval [a, b].
(a) Show that if |f(x)| ≤ |A|for all x∈[a, b], then fis constant. (A picture might help)
(b) Show that if |A|= (1/(b−a)) Rb
a|f(x)|dx, then arg fis constant.
Challenge problem: 7. Suppose fis analytic in a convex open set U. Define f0(ζ) to be the
coefficient of z−ζin the power series expansion of fbased at ζ(hence f0(ζ) = limz→ζ
f(z)−f(ζ)
z−ζ).
Suppose that for each z , w ∈Uthere exists a point ζon the line segment between zand wwith
f(z)−f(w)
z−w=f0(ζ).
Prove fis a polynomial of degree at most 2. (The point is that you have to be careful: not all
calculus theorems extend to similar complex versions.)