Exercises Complex Functions
Erik van den Ban
Spring 2007
Chapter 3
Exercise 1
Let U be an open subset of C and K a compact subset of U.
(a) Show that there exists a constant δ > 0 and finitely many points z1 , . . . , zn ∈ K such
that
K ⊂ ∪nk=1 D(zj , δ) ⊂ ∪nk=1 D(zj , 3δ) ⊂ U.
(b) Let δ be as in (a). Show that there exists a compact set K 0 ⊂ U such that, for every
z ∈ K,
D(z, δ) ⊂ K 0 .
Exercise 2
Let U be an open subset of C, let K ⊂ U be compact, and let K 0 ⊂ U, δ > 0 be as in the
previous exercise. Show that for every holomorphic function f : U → C and every p ∈ N the
following estimate is valid:
kf (p) kK ≤ p! δ −p kf kK 0 .
Hint: use the Cauchy inequalities.
Exercise 3
Let U ⊂ C be an open subset and let (fn )n≥1 be a sequence of holomorphic functions on U.
Assume that the sequence (fn ) converges to a limit function f, locally uniformly on U, i.e.,
uniformly on every compact subset of U. In the lecture we showed that f is holomorphic on
U. Use the previous exercise to show that for every p ∈ N
fn(p) → f (p) , (n → ∞),
locally uniformly on U.
Exercise 4
(a) Show that for every z ∈ C, z 6= 1,
sn (z) :=
n
X
k=0
1
zk =
1 − z n+1
.
1−z
(b) Show that the sequence sn converges locally uniformly on the unit disk D = D(0; 1),
with limit function
1
s(z) =
.
1−z
(c) Show that the following series converges locally uniformly on D :
X zk
k≥1
k
.
Hint: compare with the series in (b).
(d) Let f : D → C be the function defined by the series in (c). Show that f is holomorphic
and that
1
f 0 (z) =
.
1−z
(e) Conclude that f (z) = − log(1 − z), for |z| < 1.
Exercise 5
Let U ⊂ C be a connected open subset. Let (fn )n≥1 be a sequence of holomorphic functions
U → C and let f : U → C be holomorphic.
Assume that limn→∞ fn (z0 ) = f (z0 ) for a fixed given point z0 ∈ U, and assume that the
sequence of derivatives fn0 converges to f 0 , locally uniformly on U. The purpose of this exercise
is to prove that fn → f locally uniformly on U.
Let z1 ∈ U. Since U is open, there exists a δ > 0 such that D̄ = D̄(z1 , δ) := D(z1 , δ) ⊂ U.
Since U is connected and open, there exists a piecewise C 1 -path γ with initial point z0 and
end point z1 .
(a) Show that for every n ∈ N,
kfn − f kD̄ ≤ |fn (z0 ) − f (z0 )| + (length(γ) + δ) kfn0 − f 0 kK ,
where K = image(γ) ∪ D̄(z1 , δ).
(b) Show that fn → f, locally uniformly on U.
Exercise 6
Let U ⊂ C be an open subset, (fn )n≥0 a sequence of functions U → C, and f : U → C. Show
that the following conditions are equivalent.
(a) For every α ∈ U there exists a disc D ⊂ U centered at α such that fn → f uniformly
on D.
(b) For every compact subset K ⊂ U, fn → f uniformly on K.
(A sequence satisfying these conditions is said to be locally uniformly convergent on U.)
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Exercise 7
Let D = D(0; R) be the open disc of radius R > 0, centered at 0. Show that for every compact
subset K ⊂ D there exists a number 0 < r < R such that K ⊂ D̄(0, r).
Exercise 8
Let U be an open subset and f : U → C a holomorphic function. Assume that the set
N = {z ∈ U | f (z) = 0} is non-empty and let α ∈ U \ N.
(a) Show that there exists a point β ∈ N such that |α − w| ≥ |α − β| for all w ∈ N. The
number |α − β| is called the distance from α to N, and is denoted by d = d(α, N ).
(b) Show that the function g = 1/f is holomorphic on U \N. In particular, it is holomorphic
on the disc D = D(α, d).
(c) Show that for every n ∈ N there exists a wn ∈ D such that |g(wn )| ≥ n.
(d) Show that the power series expansion of g around the point α has a radius of convergence
equal to d.
(e) Show that the function h : z 7→ ez1+1 has a power series expansion around 0. Determine
the first three terms of this expansion. Determine its radius of convergence.
Exercise 9
Let U be a connected open subset of C containing an open interval I =] p, q [. Let coefficients
a0 , . . . , an ∈ C be given, and let f : U → C be an analytic function. Show that the following
two statements are equivalent.
(a)
n
X
k=0
ak
dk f
=0
dxk
on I;
and
(b)
n
X
k=0
ak
dk f
=0
dz k
on U.
Exercise 10
Let β ∈ C, β 6= 0 and let α ∈ C be such that αk = β.
(a) Show that there exists an open neighborhood U of β and a holomorphic function g :
U → C such that g(β) = α and g(z)k = z for all z ∈ U.
(b) Let (U, g) be a pair as in (a) and let (U1 , g1 ) be a second such pair. Show that g = g1
on an open neighborhood of β.
√
The function g in (a) is called the branch of k z locally at β that has the value α for z = β.
Exercise 11
Given an open subset U ⊂ C, we denote by O(U ) the linear space of holomorphic functions
U → C. Let ϕ : U → V be an analytic isomorphism between open subsets of C. Show that the
map ϕ∗ : f 7→ f ◦ ϕ is a linear bijection from O(V ) onto O(U ). Show that for every C 1 -curve
γ : [a, b] → U, and every f ∈ O(V ),
Z
Z
f (z) dz = (ϕ∗ f )(w) ϕ0 (w) dw.
ϕ◦γ
γ
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Exercise 12
Show that there exists a holomorphic function z 7→ w(z), defined in a neighborhood of 0, such
that
sin z 2 = w(z)2 .
Exercise 13
The purpose of this exercise is to give a proof of the local maximum modulus principle by
using the local power series expansion in a more straightforward way.
Let D be an open disc centered at 0. Moreover, let f : D → C be a non-constant holomorphic function. Then f has a power series expansion of the following form
f (z) = a0 + ak z k + ak+1 z k+1 + · · · ,
with k ≥ 1 and ak 6= 0.
(a) Show that there exists a holomorphic function g : D → C with g(0) 6= 0 and
f (z) = a0 + z k g(z),
(z ∈ D).
(b) Show that there exists a real number ϕ and a complex number ζ ∈ C such that
eiϕ a0 ≥ 0
and eiϕ ζ k g(0) = 1.
(c) Show that there exists a holomorphic function h, defined in an open neighborhood of 0,
such that h(0) = 0 and such that
eiϕ f (τ ζ) = eiϕ a0 + τ k (1 + h(τ )),
for τ in an open neighborhood of 0 in C.
(d) Show that there exists a δ > 0 such that for all τ ∈ ] 0, δ [ we have
1
1
|f (τ ζ)| ≥ |eiϕ a0 + τ k | − τ k = |a0 | + τ k .
2
2
(e) Show that |f | does not have a local maximum at 0.
Exercise 14
Let D be an open disc with center 0 and let h : U → C be a holomorphic function.
(a) Show that for all z1 , z2 ∈ D
Z
h(z2 ) − h(z1 ) =
1
h0 (z1 + t(z2 − z1 ))(z2 − z1 ) dt.
0
(b) Assume now that h0 (0) = 0. Show that there exists a R > 0 such that for all z1 , z2 ∈
D(0; R) we have
1
|h(z1 ) − h(z2 )| < |z1 − z2 |.
2
This estimate is used in the proof of the complex inverse function theorem.
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Exercise 15
The purpose of this section is to give a proof of the fundamental theorem of algebra related
to the winding number of a curve. We consider a polynomial function p : C → C of degree
n ≥ 1. Thus,
p(z) = c0 + c1 z + · · · cn z n ,
with cn 6= 0.
(a) Show that there exists a constant R > 0 such that |z| > R ⇒ p(z) 6= 0.
(b) Show that the constant R of (a) can be adapted such that, for a suitable constant C > 0,
we have
0
p (z) n 1
|z| > R ⇒ − ≤ C 2 .
p(z)
z
|z|
(c) Let γr : [0, 2π] → C be the curve given by t 7→ reit . Thus, γr parametrizes the circle of
radius r around the origin. Show that
Z 0
p (z) n
lim
−
dz = 0
r→∞ γ
p(z)
z
r
(d) Show that for all r > R we have
Z
γr
p0 (z)
dz = 2nπi.
p(z)
(e) Use (d) and Cauchy’s theorem to show that p must have a zero in C.
(f) Let σr be the curve t 7→ p(γr (t)). Show that for r > R we have
W (σr , 0) = n.
Exercise 16
Let U, V ⊂ C be open, and let ϕ : U → V an analytic isomorphism; i.e., ϕ is bijective and
both ϕ and its inverse ϕ−1 are holomorphic. Let γ be a closed C 1 -curve in U ; then ϕ ◦ γ is a
closed C 1 -curve in V. Before addressing the following problems, do Exercise 11.
(a) Prove: if γ is homologous to 0 in U, then ϕ ◦ γ is homologous to 0 in V.
(b) Assume that γ is homologous to 0 in U, and let α ∈ U \ image(γ). Prove that:
W (ϕ ◦ γ, ϕ(α)) = W (γ, α).
Exercise 17
Let a ≥ 0 be a nonnegative real number and consider the function f : C \ {±i} → C given by
f (z) =
eiaz
.
z2 + 1
For R > 1 let γR be the closed chain consisting of the real interval IR = [−R, R] and the
half circle CR given by |z| = R, Im z ≥ 0. Assume that γR is given the counterclockwise
orientation.
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(a) Give the winding numbers W (γR , i) and W (γR , −i).
(b) Compute the integral
Z
f (z) dz.
γR
(c) Show that there exists a constant C > 0 such that
Z
|
f (z) dz| ≤ C/R
CR
for all R ≥ 2.
(d) Show that
Z
∞
−∞
eiat
dt = lim
R→∞
1 + t2
Z
f (z) dz,
IR
and compute the value of the integral on the left.
(e) Determine
Z
∞
−∞
cos at
dt.
1 + t2
Chapter 5
Exercise 18, see Exercise V.3.9, Lang, p. 171
Let f be a meromorphic function on C. This means that there exists a discrete subset S of
C such that f is holomorphic on C \ S and such that f has finite order in every point of S.
By definition, discreteness of S means that for every s ∈ S there exists a δs > 0 such that
D(s, δs ) ∩ S = {s}. We assume that
lim |f (z)| = ∞.
|z|→∞
z6=S
In other words, for every M > 0 there exists a R > 0 such that for all z ∈ C \ S we have
|z| > R ⇒ |f (z)| > M.
(a) Show that there exists a constant R0 > 0 such that the function g : z 7→ 1/f (z) extends
to a holomorphic function on the open set |z| > R0 .
(b) Show that the function h : z 7→ g(1/z) has a removable singularity at z = 0 and that in
fact h(0) = 0.
(c) Show that there exists a constant N > 0 such that the function z −N f (z) is bounded on
|z| ≥ R0 + 1.
(d) Show that the function f has at most finitely many poles.
(e) Show that there exists a polynomial function q such that all singularities of qf are
removable. Thus, we may view qf as a holomorphic function.
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(f) Show that there exist constants C > 0 and n ∈ N such that |q(z)f (z)| ≤ C|z|n for
|z| ≥ R0 + 1.
(g) Use a result from Lang’s book (which one?) to conclude that qf is a polynomial function.
Conclude that f is a rational function.
Chapter VII
Exercise 19
b := C ∪ {∞} the extended complex plane (the Riemann sphere). We define
We denote by C
b by
the subsets O1 , O2 ⊂ C
b \ {∞} = C
O1 = C
and
b \ {0}.
O2 = C
Moreover, we define the bijections χ1 : O1 → C and χ2 : O2 → C by
χ1 (z) = z
and
χ2 (z) =
1
;
z
in the last formula we mean that χ2 (∞) = 0.
b is said to be open if χj (U ∩ Oj ) is open for each j = 1, 2. Accordingly,
A subset U ⊂ C
b → C
b is said to be continuous at a ∈ C
b if for every open neighborhood V of
a map f : C
b
b such that f (U ) ⊂ V. Such a map is
f (a) in C there exists an open neighborhood U of a in C
b and all µ, ν = 1, 2 such that a ∈ Oµ and f (a) ∈ Oν the
called holomorphic if for every a ∈ C
−1
function χν ◦ f ◦ χµ is complex differentiable in a neighborhood of a.
b→C
b is holomorphic, then so is the composition g ◦ f.
(a) Show: if f, g : C
b→C
b defined by J(z) = 1/z for z ∈
(b) Show that the map J : C
/ {0, ∞} and by J(0) = ∞
and J(∞) = 0 is holomorphic.
b→C
b defined by Tc (z) = z + b for z ∈ C and
(c) Show that for every b ∈ C the map Tc : C
by tb (∞) = ∞ is holomorphic.
(d) Let a ∈ C \ {0}. Show that the map Ma defined by Ma (z) = az for z ∈ C and by
Ma (∞) = ∞ is holomorphic.
(e) Show that for every M ∈ GL(2, C) the associated fractional linear transformation FM :
b→C
b is holomorphic.
C
b the collection of bijections f : C
b→C
b such that both f and its inverse
We denote by Aut(C)
f −1 are holomorphic. This collection, equipped with composition, is a group.
b
(f) Show that every fractional linear transformation belongs to Aut(C).
We denote the collection of fractional linear transformations of Ĉ by FL(Ĉ). Then FL(Ĉ) is a
subgroup of Aut(C). Moreover, the map M 7→ FM is a group homomorphism from GL(2, C)
onto FL(Ĉ). In the rest of this exercise we will show that in fact FL(Ĉ) = Aut(Ĉ). For this
purpose, assume that F ∈ Aut(Ĉ) is arbitrary.
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(g) Show that there exists a fractional linear transformation FM such that FM (∞) = F (∞).
−1
b and that G(∞) = ∞.
Show that G := FM
◦ F belongs to Aut(C)
(h) Show that G|C is a holomorphic automorphism from C onto C. Refer to an earlier
exercise to conclude that there exist a ∈ C \ {0} and b ∈ C such that G(z) = az + b for
all z ∈ C. Show that G = FN for a suitable N ∈ GL(2, C). Conclude that F = FM N ∈
b
FL(C).
b is a surjective group homomor(i) Show that the map F : M 7→ FM , GL(2, C) → Aut(C)
phism with kernel
ker F = F−1 ({I}) = {λI | λ ∈ C \ {0}}.
b Show that
Show that F(SL(2, C)) = Aut(C).
b ' SL(2, C)/{I, −I}.
Aut(C)
Exercise 20
We consider the matrix
M=
1 −i
1
i
.
(a) Give the formula for the associated fractional linear transformation Φ = FM .
(b) Show that for every x ∈ R, |Φ(x)| = 1.
(c) Use a general property of fractional linear maps to conclude that
Φ(R) = ∂D \ {1}.
Here D denotes the unit disk {z ∈ C | |z| < 1}.
b \ ∂D.
(d) Let H denote the open upper half plane. Show that Φ(H) ⊂ C
(e) Calculate F (i) and use connectedness to conclude that Φ(H) ⊂ D.
b \ D̄.
(f) Let H− denote the open lower halfplane in C. Show that Φ(H− ) ⊂ C
(g) Show that
Φ(H) = D
and
b \ D̄.
Φ(H− ) = C
(h) Conclude that Φ is an analytic diffeomorphism from D onto H. Show that the map
F 7→ Φ−1 ◦ F ◦ Φ
is a bijection (in fact a group isomorphism) from Aut(D) onto Aut(H).
(i) Show that Aut(H) consists of fractional linear transformations.
We elaborate a bit more on this. Let SU(1, 1) denote the group of matrices of the form
a b̄
b ā
with a, b ∈ C such that |a|2 − |b|2 .
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(j) Show that A 7→ FA |D defines a surjective map from SU(1, 1) onto Aut(D).
(k) Show that
M −1 SU(1, 1)M = SL(2, R).
(l) Show that for every B ∈ SL(2, R) the fractional linear transformation FB restricts to
an automorphism of H and that the map B 7→ FB |H is surjective from SL(2, R) onto
Aut(H).
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