Mathematics 414 2003–04
Exercises 2
[Due Tuesday November 18th, 2003.]
1. Suppose S ⊆ C is any set and fn : S → C (n = 1, 2, 3, . . .) are a sequence of continuous
functions. Suppose fn → f uniformly on S (as n → ∞) for a function f : S → C. Suppose
γ: [a, b] → S is a piecewise C 1 curve in S. Show that
Z
Z
lim
fn (z) dz = f (z) dz
n→∞
γ
γ
2. Suppose G ⊆ C is an open set and φ: [a, b] × G → C is a continuous function (on [a, b] ×
C ⊆ R × C ≡ R3 ). Define a function f : G → C by
Z b
f (z) =
φ(t, z) dt
(z ∈ G).
t=a
Show that f is continuous on G. [Hint. Fix z ∈ G and show how to choose δ0 > 0 so that
D(z, δ0 ) ⊆ G. Use uniform continuity of φ on the compact set [a, b] × D(z, δ0 ).]
(This fact is used in the proof of Theorem 1.20, to show that Indγ is a continuous function
on C \ γ when γ is a piecewise C 1 curve. We can write
n
1 X
Indγ (z) =
2πi j=1
Z
aj
aj−1
γ 0 (t)
dt
γ(t) − z
where γ: [a, b] → C, a = a0 < a1 < · · · < an = b and γ is C 1 on each interval [aj−1 , aj ].
Our exercise shows that each of the integrals is a continuous function of z. )
3. Suppose G ⊆ C is an open set and fn : G → C (n = 1, 2, 3 . . .) are a sequence of analytic
functions which converge uniformly on G to a function f : G → C.
(a) Show that f must be analytic. [Hint. Morera’s Theorem.]
(b) Show that
lim fn0 (z) = f 0 (z)
n→∞
for each z ∈ G. [Hint. Fix z and choose δ0 as in question 2. Use the Cauchy integral
formula to express fn0 (z) as an integral around |ζ − z| = δ0 . Apply question 1.]
4. Suppose f : C → C is an entire function and it satisfies
|f (z)| ≤ C(|z| + 1)n
(all z ∈ C),
for some constant C ≥ 0 and some nonnegative n ∈ Z. Show that f must be a polynomial
of degree at most n. [Hint: Consider the proof of Liouville’s theorem.]
5. Suppose f : C → C is an entire function which is not constant and let w ∈ C, δ > 0. Show
that there is some z ∈ C with |f (z) − w| < δ. [Hint. If not, apply Liouville’s theorem to
g(z) = 1/(f (z) − w).]
(Thus fact is called the Casorati-Weierstrass theorem. Notice that it says that f (C) is dense
in C for any nonconstant entire function f .)
6. Let G ⊆ C be open and let γ be a piecewise C 1 curve in G which is null-homotopic in G.
(a) Show that Indγ (a) = 0 for each a ∈ C \ G.
(b) Let a ∈ G \ γ and let f : G → C. Show that the Cauchy integral formula
Z
m!
f (z)
dz = f (m) (a)Indγ (a)
m
2πi γ (z − a)
holds. [Hint: Let σ be the circle |z − a| = r traversed −Indγ (a) where r > 0 is
suitably small so that σ ⊂ G. Consider the chain Γ = γ + σ in G \ {a}.]
2