Math 4023 Complex Analysis L1 Course Outline – Spring 2021 Instructor Dr. Kin Y. Li Contact Details: Rm 3471; phone 2358-7420; e-mail: makyli@ust.hk Office Hour: Monday 4:30pm-5:20pm (or by ZOOM appointment) Teaching Assistants T1A FANG Zhuyue zfangaf@connect.ust.hk T1B HO Ka Ho khhoal@connect.ust.hk Meeting Time and Venue Lectures: L1: Mondays 1:30pm-2:50pm, Fridays 9:00am-10:20am via ZOOM Tutorials: T1A: Friday 6:00pm-6:50pm via ZOOM T1B: Friday 11:00am-11:50am via ZOOM Course Description Duration: Feb. 1 to May 8 Credits: 3 units for passing the course Prerequisite: Math 2033, Math 2043 or Math 3033 This course is for learning complex analysis, which is a useful part of mathematics that deals with problems involving complex numbers. It is a course that prepares students to learn more about the powerful concepts and facts that can allow us to solve problems more cleverly. Assessment Scheme Homeworks (30%), Midterm (30%), Final Exam (40%). All records of marks or grades will be put on Canvas. Student Learning Resources Lecture Notes and Presentation Projects: Lecture notes (and/or presentation projects) may be downloaded from Canvas. Intended Learning Outcomes The School of Science Intended Learning Outcomes and the Math Department Intended Learning Out- comes will be discribed in the first class. Upon the end of the course, you should have opportunities to and should be able to ILO A: recognize the power of abstraction and generalization, and apply logical reasoning to investigate mathematical work with independent judgement (cf Science ILO 1, Math ILO 3P) ILO B: communicate effectively about math to peer and teaching staffs using available equipments or presentation softwares (cf Science ILO 4, Math ILO 4) ILO C: apply rigorous deductive reasoning to analyze and solve problems related to math profession (cf Science ILO 6, Math ILO 2) Teaching Approach There will be lectures by the instructor on materials presented in the lecture notes with emphasis on proofs. By understanding proofs and doing homeworks, you will gradually attain the ability for ILO A. Also, there will be tutorials run by the teaching assistants. In these tutorials, the teaching assistances may do more examples or assess your written works or oral presentations. For possible presentation project, you may do it via ZOOM. These will train you for ILO B. For ILO C, there will be written assignments and tests. Tentative Course Schedule (2 hours) Syllabus. Complex Sequences and Series (3 hours) Set Descriptions and Terminologies (3 hours) Continuity and Uniform Convergence (3 hours) Stereographic Projection (3 hours) Power Series. Cauchy-Riemann Equations (4 hours) Conformal Mappings, Contour Integrals, Cauchy Theory (4 hours) Harmonic Functions and Conjugates (4 hours) Morera’s Theorem, Isolated Singularities (4 hours) Residue Theory, Infinite Products. Lecture Notes for Math 4023 (Complex Analysis) c© Department of Mathematics, HKUST 1. Complex Sequences and Series Let N,Z,R denote the set of all positive integers, integers, real numbers respectively. For (x, y) ∈ R2, we will also write it as x + iy, where i = (0, 1) is to be defined as square root of −1. The set of all complex numbers is C = {x+ iy:x, y ∈ R} with addition, subtraction, multiplication and division defined as follows. For z = (x, y) = x+ iy, the real part, imaginary part, conjugate and absolute value of z are Re z = x, Im z = y, z = x− iy and |z| = √ x2 + y2 respectively. For z = x+ iy and w = u+ iv, define z + w = (x+ u) + i(y + v), z − w = (x− u) + i(y − v), zw = (xu− yv) + i(xv + yu). Note i2 = (0 + i)(0 + i) = −1 and ww = u2 + v2 = |w|2. So for w 6= 0, we may define z/w = zw/|w|2. Next, the distance between z and w is given by |z − w|. For z = x+ iy 6= 0, the argument of z, denoted by arg z, is the angle from the positive x-axis to the ray from (0, 0) to (x, y) in radian (modulo 2pi). For example, arg i is any of the numbers pi/2+2npi, where n ∈ Z . The expressions cis θ or eiθ denote cos θ+ i sin θ. Every nonzero complex number z has a polar representation z = r cis θ, where r = |z|, θ = arg z. Let z, w be complex numbers and n be an integer. The following properties can easily be verified: (1) z + z = 2 Re z, z − z = 2i Im z, zz = |z|2, z/z = cis(2 arg z) for z 6= 0, (2) z + w = z + w, z − w = z − w, zw = z w, z/w = z/w for w 6= 0, (3) |z +w| ≤ |z|+ |w|, |z − w| ≥ ∣∣|z| − |w|∣∣, |zw| = |z||w|, |z/w| = |z|/|w| for w 6= 0, (4) for z, w 6= 0, arg(zw) = arg z + argw, arg( z w ) = arg z − argw, arg(zn) = n arg z. In analysis, reasoning involving limits are very common and important. We will begin with the concept of the limit of a sequence. (For convenience, we will abbreviate “if and only if” by “iff” or “ ⇐⇒ ” in the sequel.) Definition.A sequence {z1, z2, z3, . . .} (or in short, {zn}) converges to z ∈ C (denoted by zn → z) iff for each ε > 0, there exists Nε ∈ N such that n ≥ Nε implies |zn − z| ≤ ε (in short, lim n→∞ |zn − z| = 0.) Otherwise, the sequence is said to diverge. Examples. (1) zn = zn  converges to 0 if |z| < 1,converges to 1 if z = 1,diverges otherwise. (For |z| < 1, |zn−0| = |z|n→ 0 as n→∞. For z = 1, it is clear. For |z| ≥ 1 and z 6= 1, assume zn → w. Let ε = |z − 1|/2 > 0. Then there exists N ∈ N such that n ≥ N implies |zn−w| < ε. Now 2ε = |z− 1| ≤ |z|n|z− 1| = |zn+1− zn| ≤ |zn+1−w|+ |w− zn| < 2ε, which is a contradiction.) (2) lim n→∞ n n + z = 1, since ∣∣∣∣ nn+ z − 1 ∣∣∣∣ = |z||n+ z| = √ x2 + y2√ (n + x)2 + y2 → 0 as n→∞. Often the limit of a sequence is difficult or impossible to find. We now introduce a criterion that allows us to conclude a sequence is convergent without having to identify the limit explicitly. From math analysis course, we have the following 1 Definition. A sequence {zn} in C is a Cauchy sequence iff for each ε > 0, there exists Nε ∈ N such that m,n ≥ Nε implies |zm − zn| ≤ ε. Theorem. A sequence {zn} converges in C iff {Re zn}, {Imzn} converge in R iff {zn} is a Cauchy sequence. Proof. The case zn’s are real is proved in math analysis course. For the case zn = xn + iyn’s are complex, suppose {zn} converges to z = x + iy. Since |xn − x|, |yn − y| ≤ √|xn − x|2 + |yn − y|2 = |zn − z|, by the sandwich theorem, {xn}, {yn} converge to x, y respectively. Next suppose {xn}, {yn} converge (hence are Cauchy sequences). As |zn−zm|= √|xn − xm|2 + |yn − ym|2 ≤ |xn − xm|+ |yn − ym|, it follows {zn} is a Cauchy sequence. Finally, suppose {zn} is a Cauchy sequence. Since |xn − xm|, |yn − ym| ≤ |zn − zm|, both {xn}, {yn} are Cauchy sequences. Then {xn}, {yn} converge to some x, y respectively. Since |(xn + iyn) − (x + iy)| ≤ |xn − x|+ |yn − y|, by the sandwich theorem, {zn} converges to x+ iy. QED Definitions.A series ∞∑ k=1 zk = z1 + z2 + z3 + . . . converges iff the sequence z1, z1+z2, z1+z2+z3, . . . converges (iff Sn = z1 + z2 + . . .+ zn is a Cauchy sequence). Otherwise, the series is said to diverge. There are two simple tests for checking convergence of series, namely the term test and the absolute convergence test. The former provides a necessary condition for convergence and the latter provides a sufficient condition for convergence. Term Test. If ∞∑ k=1 zk converges, then lim n→∞ zn = limn→∞(Sn − Sn−1) = limn→∞Sn − limn→∞Sn−1 = 0. Absolute Convergence Test. If ∞∑ k=1 |zk| converges, then ∞∑ k=1 zk converges (because Tn = |z1|+|z2|+. . .+|zn| is a Cauchy sequence and for m > n, |Sm − Sn| = |zn+1 + zn+2 + . . .+ zm| ≤ |zn+1|+ |zn+2|+ . . .+ |zm| = Tm − Tn = |Tm − Tn|, forcing Sn to be a Cauchy sequence.) Examples. (1) ∞∑ k=1 1 k2 + i converges because ∞∑ k=1 ∣∣∣∣ 1k2 + i ∣∣∣∣ = ∞∑ k=1 1√ k4 + 1 ( ≤ ∞∑ k=1 1 k2 ) converges. (2) ∞∑ k=1 1 k + i diverges because Re ( ∞∑ k=1 1 k + i ) = Re ( ∞∑ k=1 k − i k2 + 1 ) = ∞∑ k=1 k k2 + 1 ( ≥ ∞∑ k=1 1 2k ) diverges. Exercises 1. For what complex values z will the following series converge (a) ∞∑ n=0 ( z 1 + z )n ; (b) ∞∑ n=1 nzn; (c) ∞∑ n=0 zn 1 + z2n ? 2. When will equality occur in the triangle inequality |w+ z| ≤ |w|+ |z|? That is, under what conditions on w and z will |w + z| = |w|+ |z|? 2 3. Establish the identity ∣∣∣∣∣ n∑ k=1 αkβk ∣∣∣∣∣ 2 = n∑ k=1 |αk|2 n∑ k=1 |βk|2 − ∑ 1≤k


































































































































































































































|αkβj − αjβk|2 for the case n = 2.
(This implies the Cauchy-Schwarz inequality
∣∣∣∣∣
n∑
k=1
αkβk
∣∣∣∣∣ ≤
√√√√ n∑
k=1
|αk|2
√√√√ n∑
k=1
|βk|2.)
4. Suppose 0 < a0 ≤ a1 ≤ · · · ≤ an. Prove that the polynomial P (z) = a0zn + a1zn−1 + · · ·+ an has no
root in the open unit disk D = {z : z ∈ C, |z| < 1}. [Hint: Consider (1− z)P (z).]
5. Prove that if 11z10+10iz9+10iz− 11 = 0, then |z| = 1. [Hint: Solve for z9.] (This problem came from
the 1989 William Lowell Putnam Mathematical Competition.)
6. Let P (z) = zn + c1zn−1 + · · ·+ cn with c1, · · · , cn real. Suppose |P (i)| < 1. Prove that there is a root
x+ iy of P (z) satisfying (x2 + y2 + 1)2− 4y2 < 1. [Hint: Show P (z) must have a pair of complex roots
x± iy with y 6= 0.] (This problem came from the 1989 USA Mathematical Olympiad.)
3
2. Set Descriptions and Terminologies
The following are two sets we will used very often. For a, b ∈ C, the line segment L from a to b is consisted
of all z = a+ t(b−a) with 0 ≤ t ≤ 1, so L = {z: (z−a)/(b−a) ∈ [0, 1]}. The upper semi-circle S with center
c ∈ C and radius r is consisted of all z = c+ reiθ with 0 ≤ θ ≤ pi, so S = {z: |z − c| = r, arg(z − c) ∈ [0, pi]}.
The following are some sets we will use occasionally.
_
Imaginary
axis
Real
axis
1 1
4
1
4
)(
(1,1)
x
y
x,yz=
x
y
y=x²
i
i
x
O
S
y=-x
y
-i
O
O
Examples. (1) S = {z: |z + i| ≤ |z − 1|} consists of complex numbers that are closer to −i than 1, i.e. the
closed half plane below y = −x.
(2) To describe the parabola y = x2 in complex variable, we set x =
z + z
2
, y =
z − z
2i
and get
z − z
2i
=(
z + z
2
)2
.
(3)
{
z: 0 < arg(z − 1− i) < pi
2
}
is the open quarter-plane lying in the first quadrant with vertex at (1, 1)
and edges parallel to the x, y axes.
Definitions. (1) The set C(z0, r) = {z: |z − z0| = r} is the circle centered at z0 of radius r. The set
B(z0, r) = {z: |z − z0| < r} is the open disk centered at z0 of radius r. The set B(z0, r) = {z: |z− z0| ≤ r} is
the closed disk centered at z0 of radius r. Sometimes, B(z0, r) is also called the r-neighborhood of z0.
(2) The boundary of a set S is denoted by ∂S and consists of all points z such that every neighborhood of z
contains a point in S and a point not in S. In particular, ∂S = ∂(C \S).
(3) A set V is open iff V is a union of open disks, equivalently V contains no boundary point. A set W is
closed iff W contains all boundary points, equivalently C \W is open. A set containing some, but not all,
boundary points is neither open nor closed. The interior of a set S is S◦ = S \ ∂S and the closure of S is
S = S ∪ ∂S.
(4) A set is bounded iff it is a subset of B(0, r) for some r > 0. A set is compact iff it is closed and bounded.
(5) A set S is disconnected iff it is contained in the union of two disjoint, open sets A,B each of which
contains at least one point of S, i.e. there are open sets A and B such that S ⊆ A∪B,A∩B = ∅, S ∩A 6= ∅
and S ∩B 6= ∅. A set is connected iff it is not disconnected. A set is a region or a domain iff it is open and
connected.
Remarks. (1) V is open iff for every z ∈ V, there exists r > 0 such that B(z, r) ⊆ V.
(2)W is closed (iff C \W is open) iff zn ∈ W, zn → z imply z ∈W (since z ∈ C \W implies there is ε > 0 such
that some zn ∈ W is in B(z, ε) ⊆ C \W, contradiction; conversely if z ∈ ∂W, there is zn ∈ W in B(z, 1/n),
then zn → z ∈W .)
4
Theorem. Every region S is polygonally connected, i.e. any two points in S can be joined by a polygonal
path lying totally in S. In fact, the polygonal path can be chosen to consist of horizontal or vertical segments.
Proof. Let p ∈ S. Define A = {z ∈ S: z is connected to p by horizontal or vertical segments contained in S}
and B = S \ A. Then p ∈ A and A must be open because for each z ∈ A, there is B(z, r) ⊆ S and every
point in B(z, r) can be connected to z (hence to p) by a horizontal and a vertical segment, i.e. B(z, r) ⊆ A.
Similarly B is open because for each w ∈ B, there is B(w, r′) ⊆ S and every point in this B(w, r′) can be
connected to w (hence not to p) by a horizontal and a vertical segment, i.e. B(w, r′) ⊆ B.
Since S is connected, S = A ∪ B, A ∩ B = ∅ and A is nonempty open, B must be empty. Therefore
A = S. QED
The theorem requires S open! A circle is connected, but not polygonally connected. In advanced courses,
it will be showed that a polygonally connected set is connected. So an open and polygonally connected set
is a region.
y
x
y
x
1
Examples. (1) Let S = B(0, 1) \ {0}. Then ∂S = C(0, 1) ∪ {0} (0 is a boundary point because every
neighborhood of 0 contains points in S and 0, which is not in S). So S is open, bounded, connected, not
closed and not compact.
(2) Let S = {z: Re z ≥ 0}, the right half plane. Then ∂S is the imaginary axis. So S is closed, connected,
not open, not bounded and not compact.
(3) Let S = C(0, 1), the unit circle. Then ∂S = S. So S is closed, bounded, compact, connected and not
open.
(4) Let S =
{
z: |z| = 1
n
for n = 1, 2, 3, . . .
}
. Observe that 0 is not in S, but it is a boundary point of S. So
∂S = S ∪{0}. S is not connected because for instance, S ⊆
{
z: |z| > 3
4
}
︸ ︷︷ ︸
open
∪
{
z: |z| < 3
4
}
︸ ︷︷ ︸
open
, where the two sets
are disjoint.
Definitions. (1) A right-angled polygon is a polygon (without self intersections)
whose edges consist of horizontal and vertical segments.
(2) A simply connected region is a region such that if a right-angled polygon is in
the region, then the interior of the polygon is also in the region.
Remarks. Simply connected regions have no “holes”. A region is simply connected iff every polygon can
be “shrunk” to a point without leaving the region.
Examples. (1) The following are examples of simply connected regions: (i) open convex sets (in particular,
disks, half-planes, infinite strips), (ii) complex plane with nonintersecting infinite slits removed, (iii) open
unit disk with a spiral joining the center to the boundary removed.
o o
o
o
5
(2) The following are examples of non-simply connected regions: (i) punctured disk, (ii) annulus, (iii)
complement of a segment in C.
o o
Exercises
1. Describe the sets whose points satisfy the following relations
(a)
∣∣∣∣z − 1z + 1
∣∣∣∣ = 2; (b) |z + 1| − |z − 1| < 2;
(c) |z2 − 1| = 1; (d) arg z − 1
z + i
=
pi
3
.
2. Let α, β, γ be distinct complex numbers. Show that they are collinear if and only if Im(αβ+βγ+γα) = 0.
3. Let α, β, γ be distinct complex numbers. Show that they are the vertices of an equilateral triangle if
and only if α+ ωβ + ω2γ = 0, where ω 6= 1 is a cube root of unity.
4. Prove that if z1, z2, z3 are distinct complex numbers and |z1| = |z2| = |z3|, then arg z2 − z3
z1 − z3 =
1
2
arg
z2
z1
.
5. What is the boundary of the set {z : Re z and Im z are rational} ?
6. Let a, b, c be distinct complex numbers. Prove that a, b, c are the vertices of an equilateral triangle in C
if and only if a2 + b2 + c2 = bc+ ca+ ab.
6
3. Continuity and Uniform Convergence
Definitions. A function f :D ⊆ C → C has a limit c at z ∈ D = D ∪ ∂D (i.e. lim
w→z f(w) = c) iff for each
ε > 0, there exists δ > 0 such that 0 < |w− z| < δ implies |f(w)− c| < ε iff for every sequence zn ∈ D → z,
we have f(zn) → c. We say f is continuous at z ∈ D iff lim
w→z f(w) = f(z); f is continuous on D iff f is
continuous at every point of D.
The ε-δ definition of limit at z can be rephrased as “for each ε-neighborhood B(c, ε) of c, there is a
δ-neighborhood B(z, δ) of z such that w ∈ B(z, δ) \ {z} implies f(w) ∈ B(c, ε).” Later we will discuss about
“neighborhoods of ∞” and use that concept to extend the notion of limit to the case at ∞. Also, the limits
of sum, difference, product, quotient equal the sum, difference, product, quotient of limits can be proved
as in the real variable case, as well as the sum, difference, product, quotient and composition of continuous
functions are continuous.
Examples. (1) Polynomials P (z) = anzn + an−1zn−1 + . . .+ a0 are continuous everywhere on C.
(2) Rational functions
P (z)
Q(z)
(where P and Q are polynomials) are continuous at z, where Q(z) 6= 0.
(3) If P (x, y) =
N∑
m,n=1
amnx
myn, then f(z) = P (z, z) is continuous everywhere on C.
For functions defined by taking limits or summations, to check continuity we usually rely on uniform
convergence.
Definitions. For a function f : A→ C, the sup-norm of f on A is ‖f‖A = sup{|f(x)| : x ∈ A}. A sequence
of functions Sn:D ⊆ C → C converges uniformly to a function S:D → C iff lim
n→∞ ‖Sn − S‖D = 0, more
precisely for every ε > 0, there exists Nε ∈ N such that m ≥ Nε implies |S(z) − Sn(z)| < ε for all z ∈ D.
(For series
∞∑
k=1
fk(z), consider Sn =
n∑
k=1
fk(z).)
Weierstrass M-test. If for each k, there is a number Mk such that |fk(z)| ≤Mk for all z in D and
∞∑
k=1
Mk
converges, then
∞∑
k=1
fk(z) converges absolutely and uniformly on D.
Proof. Absolute convergence follows by comparing
∞∑
k=1
|fk(z)| with
∞∑
k=1
Mk. For uniform convergence, let
ε > 0. Since
∞∑
k=1
Mk converges, there exists Nε ∈ N such that n ≥ Nε implies
∞∑
k=n+1
Mk < ε. Then for every
z in D, ∣∣∣∣∣
∞∑
k=1
fk(z) −
n∑
k=1
fk(z)
∣∣∣∣∣ =
∣∣∣∣∣
∞∑
k=n+1
fk(z)
∣∣∣∣∣ ≤
∞∑
k=n+1
|fk(z)| ≤
∞∑
k=n+1
Mk < ε.
QED
7
Properties of Uniform Convergence.
(1) If Sk’s are continuous on D and Sk converges uniformly to S on D, then S is continuous on D and
for all z ∈ D, lim
w→z limk→∞
Sk(w) = lim
w→z S(w) = S(z) = limk→∞
Sk(z) = lim
k→∞
lim
w→z Sk(w). (For series, if fk’s are
continuous on D and
∞∑
k=1
fk converges uniformly to f on D, then f is continuous on D and for all z ∈ D,
lim
w→z
∞∑
k=1
fk(w) = lim
w→z f(w) = f(z) =
∞∑
k=1
fk(z) =
∞∑
k=1
lim
w→z fk(w).)
(2) If Sk’s are integrable and converges uniformly on [a, b], then
∫ b
a
lim
k→∞
Sk(x) dx = lim
k→∞
∫ b
a
Sk(x) dx. (Also,
if fk’s are integrable and
∞∑
k=1
fk converges uniformly on [a, b], then
∫ b
a
∞∑
k=1
fk(x) dx =
∞∑
k=1
∫ b
a
fk(x) dx.)
Proof. (1) Given ε > 0. Since Sk converges uniformly to S on D, there exists N = Nε/3 such that k ≥ Nε/3
implies |S(z) − Sk(z)| < ε3 for all z ∈ D. For a fixed w ∈ D, since SN is continuous at w, there exists δ > 0
such that |w− z| < δ ⇒ |SN (w)− SN (z)| < ε3 . Then
|S(w)− S(z)| = |S(w) − SN (w) + SN (w)− SN (z) + SN (z) − S(z)|
≤ |S(w) − SN (w)|+ |SN (w)− SN (z)|+ |SN (z) − S(z)| < ε3 +
ε
3
+
ε
3
= ε.
So S is continuous onD. The parenthetical statement follows from the first statement by taking Sn =
n∑
k=1
fk.
(2) Let εn = ‖S−Sn‖[a,b]. Then |S(x)−Sn(x)| ≤ εn, which is equivalent to Sn(x)− εn ≤ S(x) ≤ Sn(x)+ εn
for all x ∈ [a, b]. So∫ b
a
Sn(x) dx− εn(b− a) = (L)
∫ b
a
(Sn(x)− εn) dx ≤ (L)
∫ b
a
S(x) dx
≤ (U )
∫ b
a
S(x) dx ≤ (U )
∫ b
a
(Sn(x) + εn) dx =
∫ b
a
Sn(x) dx+ εn(b− a).
(†)
Hence, 0 ≤ (U )
∫ b
a
S(x) dx− (L)
∫ b
a
S(x) dx ≤ 2εn(b− a). Since lim
n→∞ εn = 0, by sandwich theorem, we get
(U )
∫ b
a
S(x)dx = (L)
∫ b
a
S(x)dx, i.e. S(x) is integrable on [a, b].By (†),
∣∣∣∣∣
∫ b
a
Sn(x) dx−
∫ b
a
S(x) dx
∣∣∣∣∣ ≤ εn(b − a).
Letting εn → 0, lim
n→∞
∫ b
a
Sn(x) dx =
∫ b
a
S(x) dx =
∫ b
a
lim
n→∞Sn(x) dx. The series case follows as well. QED
It is not true that a convergent series of continuous functions is continuous. The following is an
Example. For k ∈ N, fk(z) = z(1 + |z|)k is continuous on C . Using the geometric series formula with
r =
1
1 + |z| , we see
∞∑
k=1
fk(z) =
∞∑
k=1
zrk =
zr
1− r =
{
0 if z = 0
z/|z| if z 6= 0 is not continuous at 0. Thus, the sum of
infinitely many continuous functions may converge on C, yet the sum function may still not be continuous.
Next, consider f(z) =
∞∑
k=0
zk
k!
. Suppose z ∈ B(0, R). For such z, we have
∣∣∣∣zkk!
∣∣∣∣ ≤ Rkk! = Mk. The series
∞∑
k=0
Rk
k!
converges by the ratio test because lim
k→∞
Rk+1
(k + 1)!
/
Rk
k!
= lim
k→∞
R
k + 1
= 0. By Weierstrass M-test,
8
∞∑
k=0
zk
k!
converges uniformly on B(0, R). Since
zk
k!
are continuous on B(0, R), by property (1) of uniform
convergence, f is continuous on B(0, R). Finally, since this is true for all R > 0, f is continuous on C .
Exercises
1. Prove that
∞∑
n=1
z
n(1 + n|z|2) converges uniformly on C .
2. Show that f(z) =
∞∑
k=1
1
k2 + z
is continuous on the right half plane H = {z : Re z > 0}.
3. Show that f(z) =
∞∑
k=0
kzk is continuous on D = {z : |z| < 1}.
4. It is known that ln(1 + x) =
∞∑
n=1
(−1)n−1xn
n
for every x ∈ [0, 1]. Prove that
∞∑
n=1
(
x2n+1
2n+ 1
− x
n+1
2n+ 2
)
converges for every x ∈ [0, 1], but does not converge uniformly on [0, 1]. [Hint: Consider
∞∑
n=1
(
x2n − x
n
2
)
on [0, w] with w < 1.]
5. Prove that
∞∑
n=1
zn
n
does not converge uniformly on B(0, 1). [Hint: |Sn(z) − Sm(z)| < ε and z → 1.]
6. Prove that for every ε > 0,
∞∑
k=1
1
k
xk
1 + x2k
converges uniformly on [1 + ε,+∞).
9
4. Stereographic Projection
In many situations, it is best to treat infinity as a point. Stereographic projection explains how this can
be done.
Let S2 = {(u, v, w):u2 + v2 + w2 = 1} ∈ R3 be the
unit sphere and N = (0, 0, 1) be the north pole. There is
a bijection between S2 \N and the complex plane C given
by Z = (u, v, w)↔ z = x + iy = (x, y, 0) as in the figure.
Since z−N = (x, y,−1) and Z −N = (u, v, w− 1), we get
x
u
=
y
v
=
−1
w − 1 .
Then x = u/(1−w), y = v/(1−w). Also, 1 = u2+v2+w2 =
(x2+y2)(1−w)2+w2. Then x2+y2 = 1 +w
1−w =
2
1−w −1.
So
Z
O
N
z Z
z
u = x(1− w) = 2x
x2 + y2 + 1
, v = y(1 −w) = 2y
x2 + y2 + 1
, w =
x2 + y2 − 1
x2 + y2 + 1
.
If z = x+ iy →∞, then x2 + y2 = |z|2 →∞, so Z = (u, v, w)→ N = (0, 0, 1). Thus N = (0, 0, 1)↔∞.
The set C∪{∞} is usually referred to as the extended complex plane. Because of the correspondence,
some like to use S2 to denote the same set. If S2 is used, then the extended complex plane is sometimes also
referred to as the Riemann sphere.
Now to analyze how the neighborhoods of a point are transformed under stereographic projection,
we study the correspondence of the boundary circles. A circle on S2 is the intersection of S2 with a
plane Au + Bv + Cw = D. This corresponds to (C − D)(x2 + y2) + 2Ax + 2By = C + D, which is a{ circle on C if C 6= D
line on C if C = D
.
Conversely, a circle x2+ y2 + ax+ by = c on C corresponds to a circle au+ bv+ (c+ 1)w = c− 1 on S2
not containing N ; a line ax+ by = c on C correspond to a circle au+ bv + cw = c on S2 containing N.
In particular, the circle from S2 intersecting the plane w = w0 < 1 corresponds to x2 + y2 =
1 +w0
1−w0 ,
i.e. the boundary of a neighborhood of N corresponds to a circle centered at the origin. Then deleted
neighborhoods of N correspond to sets of the form {z: |z| > r}. By abuse of notation, we will let B(∞, r) =
{z: |z| > r}. The remark following the definition of lim
w→z f(w) suggests how limits involving∞ can be defined.
Definitions. (1) zn →∞ iff |zn| → ∞ iff for any ε > 0, there exists Nε ∈ N such that n ≥ Nε ⇒ |zn| > ε.
(2) lim
z→∞ f(z) = c ∈ C iff for any ε > 0, there exists δ > 0 such that |z| > δ ⇒ |f(z)−c| < ε iff limw→0 f
( 1
w
)
= c.
In that case, we define f(∞) = c and say f(z) is continuous at ∞.
(3) lim
z→a∈C
f(z)=∞ iff for any ε > 0, there exists δ > 0 such that 0 < |z−a| <δ ⇒ |f(z)| > ε iff lim
z→a
1
f(z)
= 0.
(4) lim
z→∞f(z) =∞ iff for any ε > 0, there exists δ > 0 such that |z| > δ ⇒ |f(z)| > ε.
10
Definitions. Let U be a subset of C and f : U → C be a function.
(1) f is differentiable at z iff there exists B(z, r) ⊆ U such that
f ′(z) = lim
h→0
f(z + h)− f(z)
h
∈ C, where h ∈ B(0, r).
f is differentiable at ∞ iff U contains someB(∞, r), f is continuous at∞ and g(z) =
{
f(1/z) if 0 < |z| < 1/r
f(∞) if z = 0
is differentiable at 0. In that case, we define f ′(∞) = g′(0).
(2) f is holomorphic or analytic (or regular) on a set S iff f is differentiable at every point of some open set
containing S. In case S = {z}, f is holomorphic at z iff f is differentiable at every point of some open set
(like B(z, r)) containing z.
(3) f is univalent (or schlicht) on an open set iff f is differentiable and injective there.
(4) f is entire (or integral) iff f is differentiable on the complex plane.
Of course, as usual, the sum, difference, product, quotient and chain rules are valid in (complex)
differentiation. Some common functions are differentiable, e.g. polynomials are entire functions and (zn)′ =
nzn−1 for integer n. However, some are not, e.g. the conjugate function z is not differentiable anywhere
because lim
h→0,h∈C
z + h− z
h
= lim
h→0,h∈C
h
h
doesn’t exist, as can be seen by the following computations :
lim
h→0,h∈R
h
h
= lim
h→0,h∈R
h
h
= 1 and lim
h=it→0,t∈R
h
h
= lim
t→0
−it
it
= −1.
Exercises
1. The chordal metric d(z, z′) of two complex numbers z and z′ is the distance in R3 between the points
Z and Z′ corresponding to z and z′ respectively under stereographic projection.
(a) Express d(z, z′) in terms of z and z′ only.
(b) Express d(z,∞) = lim
z′→∞
d(z, z′) in terms of z only.
(c) Show that zn → z ∈ C∪{∞} if and only if d(zn, z)→ 0.
2. Prove that f(z) = |z| is not differentiable anywhere, but g(z) = |z|2 is differentiable at 0 only.
3. Define f : C \{1} → C by f(z) = 1/(1 − z). Prove f can be extended to be continuous at ∞. Is f
differentiable at ∞?
4. Give an example of a function f(z) that is continuous on C∪{∞}, but f(z) is not differentiable at 0.
5. Prove that if z ∈ C \{0} corresponds to Z = (u, v, w) ∈ S2 under stereographic projection, then 1/z
corresponds to (u,−v,−w), which is the point Z∗ on S2 such that the midpoint of ZZ∗ is (u, 0, 0) ∈ R3 .
6. Prove that if the circle formed by S2 intersecting the plane Au+Bv +Cw = D corresponds to a circle
(or line) Γ on C under stereographic projection, then Γ∗ = {1/z : z ∈ C} corresponds to the circle
formed by S2 intersecting the plane −Au+ Bv + Cw = −D.
11
5. Power Series
Definition. A power series centered at c is a function of the form f(z) =
∞∑
n=0
an(z − c)n. (Note that
f(c) = a0, so the series converges for at least the point z = c.)
To understand what the domains of power series (as functions) look like, we introduce the concept of
the upper limit of a sequence and recall the root test.
Definition. Let {xn} be a sequence in R . The limit superior or upper limit of {xn} (denoted by limsup
n→∞
xn
or lim
n→∞xn) is sup
{
x ∈ [−∞,+∞] : x = lim
k→∞
xnk, where {xnk} is a subsequence of {xn}
}
. More precisely,
it is
(a) the number L if L has the properties that for each ε > 0,
(i) xn < L+ ε for all except finitely many n and
(ii) xn > L− ε for infinitely many n,
(b) +∞ if for each real number r, there is xn > r,
(c) −∞ if for each real number r, only finitely many xn’s are greater than r.
Essentially, by taking ε→ 0, items (i) and (ii) imply L is the limit of some subsequence. Furthermore,
item (i) implies no larger number is also a limit of a subsequence. Now if lim
n→∞xn exists, then all subsequences
converge to the same limit. In that case, limsup
n→∞
xn = lim
n→∞xn.
Examples. (1) lim
n→∞
1
n
= 0 implies limsup
n→∞
1
n
= 0.
(2) limsup
n→∞
[n+ (−1)nn] = limsup{0, 4, 0, 8,0, 12, . . .} = lim{4, 8, 12, . . .} = +∞.
(3) limsup
n→∞
sin
npi
2
= limsup{1, 0,−1, 0, 1,0, . . .} = lim{1, 1, 1, . . .} = 1.
Root Test. Given a series
∑
an, let ρ = limsup
n→∞
n
√
|an|, then
∑
an
{ converges absolutely if ρ < 1,
diverges if ρ > 1,
is inconclusive if ρ = 1.
Proof. (Case ρ < 1) Take r such that ρ < r < 1. Let ε = r − ρ > 0. Then by property (i), all but finitely
many n satisfy n
√|an| < ρ + ε = r, then |an| < rn. Since r < 1, ∑ rn converges, which implies ∑ |an|
converges.
(Case ρ > 1) Take r such that 1 < r < ρ. Let ε = ρ − r > 0. Then by property (ii) n√|an| > ρ − ε = r > 1
for infinitely many n, then lim
n→∞ |an| 6= 0, so
∑
an diverges.
(Case ρ = 1) Consider
∞∑
n=1
1
n
and
∞∑
n=1
1
n2
. For both cases, ρ = 1, but the first series diverges and the second
series converges. QED
12
Theorem 1. The power series f(z) =
∞∑
n=0
an(z − c)n converges absolutely for |z − c| < R and diverges for
|z − c| > R, where R = 1
limsup
n→∞
n
√
|an|
. (In case R = 0, the series converges at z = c only.) R is called the
radius of convergence and the disk |z− c| < R is called the disk of convergence. Furthermore,
∞∑
n=0
an(z − c)n
converges uniformly on any smaller disk |z − c| ≤ R′ < R.
Proof. The first statement follows from the root test. The last statement follows by Weierstrass M-test.
Since |an(z − c)n| ≤ |an|R′n for all |z − c| ≤ R′, so we set Mn = |an|R′n and use the absolute convergence
of the power series at z = R′ + c. QED
In math analysis, we learned that if lim
n→∞
∣∣∣∣an+1an
∣∣∣∣ exists, then limn→∞n√|an| = limn→∞
∣∣∣∣an+1an
∣∣∣∣. This can be
used in computing the radius of convergence.
The convergence on the boundary circle |z − c| = R can be arbitrary as the following examples show.
Examples. (1) For
∞∑
n=1
zn
n2
, R = 1. If |z| = 1, then
∞∑
n=1
∣∣∣∣znn2
∣∣∣∣ = ∞∑
n=1
1
n2
converges by p-test.
(2) For
∞∑
n=0
zn, R = 1. If |z| = 1, then
∞∑
n=0
zn diverges because lim
n→∞ z
n 6= 0.
(3) For
∞∑
n=1
zn
n
, R = 1. If z = 1,
∞∑
n=1
1
n
diverges. If z = −1,
∞∑
n=1
(−1)n
n
converges by alternating series test.
Later we will show that every holomorphic function can be locally represented by power series (i.e. f
holomorphic at c implies f(z) =
∞∑
n=0
an(z − c)n in a neighborhood of c). Therefore, the local properties of
holomorphic functions can be understood by studying power series. First, we will see that power series can
be differentiated (term-by-term) in their disks of convergence.
Theorem 2. If f(z) =
∞∑
n=0
an(z − c)n converges on |z − c| < R, then f ′(z) =
∞∑
n=1
nan(z − c)n−1 converges
on |z − c| < R, too. Consequently, power series are infinitely differentiable on |z − c| < R.
Proof. The proof is divided into two steps. The first step is to observe that xnk = |ank|1/nk → c ⇐⇒
nk
1/nk |ank|1/nk → c and lnxnk → ln c ⇐⇒ nknk−1 lnxnk → ln c. Hence, by the definition of limsup,
limsup
n→∞
n−1
√
n |an| = limsup
n→∞
(
(n |an|)
1
n
) n
n−1
= limsup
n→∞
(
n
1
n |an|
1
n
)
= limsup
n→∞
n
√
|an|,
so the power series of f(z) and f ′(z) have the same radius of convergence. The second step is to obtain an
inequality of the form
∣∣∣∣∣f(z + h)− f(z)h −
∞∑
n=1
nan(z − c)n−1
∣∣∣∣∣ ≤ Az|h|.
Fix z such that |z − c| < R. It suffices to consider small h (say |h| < δ < R− |z − c|.) Then
f(z + h)− f(z)
h
−
∞∑
n=1
nan(z − c)n−1 =
∞∑
n=2
an
(
n∑
k=2
(
n
k
)
hk−1(z − c)n−k
)
︸ ︷︷ ︸
bn
.
13
Observe that for k ≥ 2,(
n
k
)
=
(
n
k − 2
)
(n− k + 2)
k − 1
(n − k + 1)
k
≤
(
n
k − 2
)
n(n− 1).
So, for z 6= c,
|bn| ≤
∞∑
k=2
(
n
k
)
|h|k−1|z − c|n−k ≤ n(n− 1)|h||z − c|2
n∑
k=2
(
n
k − 2
)
|h|k−2|z − c|n−k+2
≤ n(n− 1)|h||z − c|2
n∑
j=0
(
n
j
)
|h|j|z − c|n−j = n(n− 1)|h||z − c|2 (|z − c|+ |h|)
n
.
Therefore, as h→ 0,∣∣∣∣∣f(z + h)− f(z)h −
∞∑
n=1
nan(z − c)n−1
∣∣∣∣∣ =
∣∣∣∣∣
∞∑
n=2
anbn
∣∣∣∣∣ ≤
∞∑
n=2
|an||bn|
≤ |h| 1|z − c|2
∞∑
n=2
n(n − 1)|an|(|z − c|+ δ)n︸ ︷︷ ︸
Az
→ 0.
(Note that the series in Az converges because the first step above implies f ′′(w) =
∑∞
n=2 n(n−1)an(w−c)n−2
converges absolutely on |w − c| < R, in particular at w = c+ |z − c|+ δ by the condition on δ.) QED
Taylor’s Theorem for Power Series. If f(z) =
∞∑
n=0
an(z − c)n has a nonzero radius of convergence, then
the coefficient an =
f (n)(c)
n!
, i.e. f(z) equals its Taylor series
∞∑
n=0
f (n)(c)
n!
(z − c)n.
Proof. Apply theorem 2 repeatedly (i.e. inductively) and evaluate at z = c. QED
Identity Theorem for Power Series. If f(z) =
∞∑
n=0
an(z − c)n = 0 for z = zk 6= c, (k = 1, 2, . . .) and {zk}
converges to c, then an = 0 for all n.
Proof.We are given a power series f(z) = a0+a1(z−c)+a2(z−c)2+ · · ·. Now a0 = f(c) = lim
k→∞
f(zk) = 0,
a1 = lim
z→c
f(z)
z − c = limk→∞
f(zk)
zk − c = 0, . . ., an = limz→c
f(z)
(z − c)n = limk→∞
f(zk)
(zk − c)n = 0, . . . . QED
Uniqueness Theorem for Power Series. If
∞∑
n=0
an(z− c)n =
∞∑
n=0
bn(z− c)n for z = zk 6= c, (k = 1, 2, . . . )
and {zk} converges to c, then an = bn for all n.
Proof. Let f(z) =
∞∑
n=0
(an − bn)(z − c)n. As all f(zk) = 0, by identity theorem, an − bn = 0 for all n. QED
Exercises
1. Find limsup
n→∞
xn, where
14
(a) xn = n sin
(
1
n
)
; (b) xn = cos
npi
4
;
(c) xn = 1 + (−1)n 2n
n+ 1
.
2. Find the radius of convergence for the following power series:
(a)
∞∑
n=1
zn!; (b)
∞∑
n=0
(n+ 2n)zn.
3. Give an example of two power series
∞∑
n=0
anz
n and
∞∑
n=0
bnz
n with radii of convergence R1 and R2,
respectively, such that the power series
∞∑
n=0
(an + bn)zn has a radius of convergence greater than R1+R2.
4. Show that if |z| < 1, then
∞∑
n=0
zn =
1
1− z . Find a power series for
1
(1− z)2 with centered at 0. What
is the radius of convergence for this series?
5. Explain why there is no power series f(z) =
∞∑
n=0
cnz
n such that f
(
1
k
)
= 1 for k = 2, 3, 4, . . . and
f ′(0) > 0.
6. Does there exist a power series f(z) =
∞∑
n=0
cnz
n such that f
(
1
k
)
=
1
k2
and f
(
−1
k
)
=
1
k3
for k =
1, 2, 3, . . .?
7. If f(z) =
∞∑
n=0
cnz
n satisfies f
(
1
k
)
=
k2
k2 + 1
, k = 1, 2, 3, . . . , compute the values of the derivatives
f (n)(0), n = 1, 2, 3 . . ..
15
6. Cauchy−Riemann Equations
Recall f is differentiable at z0 means lim
h→0,h∈C
f(z0 + h)− f(z0)
h
= f ′(z0) exists and f is holomorphic at
z0 means f is differentiable on an open set containing z0, in which case f ′(w) exists in some disk B(z0, r)
about z0. We now ask the
Question. Given a function f :R2 → R2 with two real variables, say f(x, y) = (u(x, y), v(x, y)) = u(x, y) +
iv(x, y), how can we tell if it is a differentiable function of z = x+ iy?
Example. If f(z) = z2, z = x+ iy, then f(x, y) = (x+ iy)2 = (x2 − y2) + i(2xy) = (x2 − y2, 2xy).
Theorem. If f(z) = f(x, y) = (
real︷ ︸︸ ︷
u(x, y),
real︷ ︸︸ ︷
v(x, y)) = u(x, y) + iv(x, y) = u(z) + iv(z) is differentiable at z0,
then
∂u
∂x
,
∂u
∂y
,
∂v
∂x
,
∂v
∂y
exist at z0,
∂u
∂x
(z0) =
∂v
∂y
(z0) and
∂u
∂y
(z0) = −∂v
∂x
(z0).
[Notation. If we define fx =
∂f
∂x
=
∂u
∂x
+ i
∂v
∂x
and fy =
∂f
∂y
=
∂u
∂y
+ i
∂v
∂y
, then the above Cauchy-Riemann
equations
∂u
∂x
=
∂v
∂y
and
∂u
∂y
= −∂v
∂x
are equivalent to fy = ifx.]
Proof. Taking h→ 0 along the real axis, we have
f ′(z0) = lim
h→0
f(x0 + iy0 + h)− f(x0 + iy0)
h
= lim
h→0
f(x0 + h, y0)− f(x0, y0)
h
= fx(x0, y0).
Taking h→ 0 along the imaginary axis, say h = it, we have
f ′(z0) = lim
t→0
f(x0 + iy0 + it)− f(x0 + iy0)
it
=
1
i
lim
t→0
f(x0, y0 + t)− f(x0, y0)
t
=
1
i
fy(x0, y0).
QED
Theorem. If fx and fy exist in a neighborhood of z0, are continuous at z0 and fy(z0) = ifx(z0), then f is
differentiable at z0. (If fx, fy are continuous and fy = ifx on B(z0, r), then f is holomorphic on B(z0, r).)
Proof. Write f(z0) = u(x0, y0) + iv(x0, y0), h = s + it. Then
u(z0 + h)− u(z0)
h
=
u(x0 + s, y0 + t) − u(x0, y0)
s + it
=
u(x0 + s, y0 + t) − u(x0 + s, y0)
s + it
+
u(x0 + s, y0)− u(x0, y0)
s + it
=
t
s+ it
[
∂u
∂y
(x0 + s, y0 + αt)
]
+
s
s + it
[
∂u
∂x
(x0 + βs, y0)
]
, 0 < α, β < 1
(by the mean value theorem).
Similarly,
v(z0 + h)− v(z0)
h
=
t
s + it
[
∂v
∂y
(x0 + s, y0 + γt)
]
+
s
s+ it
[
∂v
∂x
(x0 + δs, y0)
]
, 0 < γ, δ < 1.
16
Now, fy(z0) = ifx(z0) implies fx(z0) =
t
s + it
fy(z0) +
s
s + it
fx(z0). Using these equations and the fact∣∣∣∣ ts + it
∣∣∣∣ , ∣∣∣∣ ss + it
∣∣∣∣ ≤ 1, it follows that as h = s + it→ 0,∣∣∣∣f(z0 + h)− f(z0)h − fx(z0)
∣∣∣∣ = ∣∣∣∣ ts + it
[
∂u
∂y
(x0 + s, y0 + αt) + i
∂v
∂y
(x0 + s, y0 + γt) − fy(z0)
]
+
s
s+ it
[
∂u
∂x
(x0 + βs, y0) + i
∂v
∂x
(x0 + δs, y0)− fx(z0)
]∣∣∣∣
≤
∣∣∣∣∂u∂y (x0 + s, y0 + αt) + i∂v∂y (x0 + s, y0 + γt) − fy(z0)
∣∣∣∣ + ∣∣∣∣∂u∂x (x0 + βs, y0) + i ∂v∂x (x0 + δs, y0) − fx(z0)
∣∣∣∣→ 0.
So f is differentiable at z0 and f ′(z0) = fx(z0). QED
Examples. (1) If G(z) = u(z) + iv(z) = u(x, y) + iv(x, y) is holomorphic on a region D and has the real
part u(x, y) = x2 − y2, what can v(z) be?
Solution.
∂v
∂x
= −∂u
∂y
= 2y ⇒ v(x, y) = 2xy +C1(y)
∂v
∂y
=
∂u
∂x
= 2x⇒ v(x, y) = 2xy +C2(x)
⇒ v(x, y) = 2xy + constant.
(2) If f = u+ iv is holomorphic on a region D and u ≡ constant, show that f ≡ constant.
Solution.
∂v
∂x
= −∂u
∂y
= 0⇒ v(x, y) = C1(y)
∂v
∂y
=
∂u
∂x
= 0⇒ v(x, y) = C2(x)
⇒ v(x, y) ≡ constant⇒ f ≡ constant.
(3) If f is holomorphic on a region D and |f(z)| ≡ constant, then f ≡ constant.
Solution. If |f | ≡ 0, then f ≡ 0. Otherwise u2+v2 ≡ constant, so taking partial derivatives, we get 2u∂u
∂x
+
2v
∂v
∂x
≡ 0, 2u∂u
∂y
+ 2v
∂v
∂y
≡ 0. Since ∂u
∂y
= −∂v
∂x
,
∂v
∂y
=
∂u
∂x
, we get
∂u
∂x
(
=
∂v
∂y
)
≡ 0, ∂v
∂x
(
= −∂u
∂y
)
≡ 0.
Then u ≡ constant, v ≡ constant, so f ≡ constant.
Examples (2) and (3) are special cases of the open mapping theorem (see page 43).
Exercises
1. If f = u+ iv is holomorphic on some domain, given u(x, y) below, find the possibilities of v(x, y):
(a) u(x, y) = x3 − 3xy2 ; (b) u(x, y) = e−y cosx;
(c) u(x, y) = ln(x2 + y2); (d) u(x, y) =
y
(1− x)2 + y2 .
2. Show that there is no holomorphic function f = u+ iv with u(x, y) = x2 + y2.
3. Suppose f is an entire function of the form f(x, y) = u(x) + iv(y). Prove that f is a polynomial of
degree at most one.
17
4. Suppose that f and f are holomorphic on a region D. Show that f is a constant function on D.
5. Let G be a region and G∗ = {z : z ∈ G} be the mirror image of G across the x-axis. If f : G → C is
holomorphic, show that f∗ : G∗ → C defined by f∗(z) = f(z) is holomorphic.
6. Write z in polar coordinates as (r, θ). Then f(z) = u(z) + iv(z) = (u(r, θ), v(r, θ)). Establish the polar
form of the Cauchy-Riemann equations
∂u
∂r
=
1
r
∂v
∂θ
and
∂v
∂r
= −1
r
∂u
∂θ
.
7. Let f = u + iv be holomorphic on a region D, where u, v are real-valued functions on D. If there are
real numbers a, b, c such that a2 + b2 6= 0 and au+ bv = c in D, then prove that f is constant in D.
18
7. Definitions of Common Functions
In this section, we will enlarge our collection of complex-valued functions by extending the common
functions, such as exponential, logarithm and trigonometric functions, to complex domains.
To define ez , we want it to satisfy (1) ew+z = ewez, (2) ex is the same as before for x real and (3) ez is
entire.
Let f(z) be ez . Then we want f(w + z) = f(w)f(z) and f(x) = ex for x real. So f(z) = f(x + iy) =
exf(iy) = ex(A(y) + iB(y)). For f to be entire (differentiable everywhere), the Cauchy-Riemann equations
require fy = ifx, i.e. ex(A′(y) + iB′(y)) = iex(A(y) + iB(y)). Then A′(y) = −B(y) and B′(y) = A(y).
So A′′(y) = −A(y) and B(y) = −A′(y). Now f(0) = 1, so A(0) = 1 and B(0) = 0. Then B′(0) = 1 and
A′(0) = 0. Therefore, A(y) = cos y and B(y) = sin y. So it makes sense to have the following
Definition. For z = x+ iy, ez = ex(cos y + i sin y) = ex cis y. (In particular, eiy = cis y.)
An interesting fact is the famous Euler’s equation eipi + 1 = 0, which relates five of the most important
constants e, i, pi, 1, 0 in mathematics.
Properties. From the definition, we have the following facts:
(1) ew+z = ewez , (2) |ez| = ex, (3) ez 6= 0 for all z, (4) d
dz
ez = ez .
Since eiy = cos y + i sin y, e−iy = cos y − i sin y, so cos y = e
iy + e−iy
2
and sin y =
eiy − e−iy
2i
. We will
use these identities to define trigonometric functions of complex numbers.
Definitions. cos z =
eiz + e−iz
2
, sin z =
eiz − e−iz
2i
, tan z =
sin z
cos z
, sec z =
1
cos z
, csc z =
1
sin z
,cot z =
cos z
sin z
.
The derivatives of these functions are the same as before. Also, cos2 z + sin2 z ≡ 1 for all z and the
usual trigonometric identities are still valid. However, it is not true that | cos z| ≤ 1 and | sin z| ≤ 1 for all z,
e.g. cos 10i =
e−10 + e10
2
> 1000.
Now we turn our attention to defining logarithm. This is done by inverting exponentiation.
Definitions. If U, V are open sets, f :U → V is bijective, then the inverse of f is f−1:V → U defined by
f−1(v) = u whenever f(u) = v. Also, for f differentiable on U, we say g is the inverse of f at z ∈ V if g is
the inverse of f in some neighborhood of z in V .
Consider the equation ez = w with w 6= 0. Let z = x+iy be a solution of ez = w. Then ez = ex cis y = w,
so ex = |w|, y = argw. Hence z = ln |w| + i argw. However, argw is multivalued. Let Argw denote the
so-called “principal branch” of argument, which is defined by −pi < Argw ≤ pi. The general solution of
ez = w is z = ln |w|+ i(Argw + 2kpi), where k is any integer.
Definitions. For w 6= 0, logw = ln |w| + i argw, where argw = Argw + 2kpi, k any integer. The choice
Logw = ln |w|+ iArgw is the principal branch of the logarithm function.
To make logw continuous, we need to choose argw continuously.
19
0 1
1 20
Examples. (1) On the annulus A = {w: 1 < |w| < 2}, argw cannot be defined con-
tinuously. This can be seen by looking at the values of argw on the circle |w| = 3
2
.
At w =
3
2
, argw = 2kpi. As w moves on the circle |w| = 3
2
counterclockwise, the
value of argw increases. When it get back to w =
3
2
at the end, the value of argw
have increased by 2pi from the starting 2kpi. So argw cannot be made continuously.
(2) For the region on the left, argw can be defined continuously. We can define
argw by choosing arg 1 = Arg 1 = 0 and for any other w in the region, draw a
polygonal path joining 1 to w and define argw continuously along the path by
starting at 1. It is visually clear that argw will be well-defined (even if different
paths are used) and continuous. It is bad to simply define argw by taking the
principal branch Argw because the region “wraps” around the origin a little more
than once.
On the word “branch” If argw is defined continuously on a region, then argw+2kpi (for any fixed integer
k 6= 0) is also continuous on the same region and gives also the argument of points on the region. We refer
to both of these functions as different “branches” of the argument function.
Similarly, if log z is defined continuously on a domain, then log z +2pii is also a continuous inverse of ez
(since elog z+2pii = z = elog z). We simply say that both are different “branches” of the logarithm function.
To differentiate the logarithm function, we will apply the inverse rule “
dw
dz
= 1
/
dz
dw
”.
Inverse Rule. Let f be bijective near w0 and f(w0) = z0. Let g = f−1 be continuous near z0. If f is
differentiable at w0 = g(z0) and f ′(g(z0)) 6= 0, then g is differentiable at z0 and g′(z0) = 1
f ′(g(z0))
.
Proof. Since f(g(z)) = z in a neighborhood of z0, g(z) → g(z0) as z → z0 in that neighborhood, we have
g(z) − g(z0)
z − z0 =
1
f(g(z)) − f(g(z0))
g(z) − g(z0)
→ 1
f ′(g(z0))
due to g injective near z0 and f ′(g(z0)) 6= 0.
QED
From the theorem, we see that if argw can be defined continuously on a region, then logw will be
continuous and even differentiable. Just take g(z) = log z, f(z) = ez, then
d
dz
log z = g′(z) =
1
elog z
=
1
z
.
Definitions. For α 6= 0, define αβ = e(β logα) and the principal value of αβ is e(β Logα).
Example. ii = e(i log i) = ei(ln |i|+iarg i) = ei(0+i(
pi
2+2kpi)) = e−
pi
2−2kpi (which is amazingly all real-valued!).
The principal value of ii is e−
pi
2 .
Remarks. (1) For α 6= 0, d
dz
αz =
d
dz
e(z logα) = αz logα (for each fixed value of logα.)
(2) If arg z can be defined continuously on a region, then
d
dz
zα =
d
dz
e(α log z) = e(α log z)
α
z
= αzα−1.
Example. We will show that, on Ω = C\ ((−∞,−1]∪ [1,+∞)) , there is a continuous branch of
√
z2 − 1.
Observe that √
z2 − 1 = (z2 − 1)1/2 = e 12 log(z2−1) = e 12 (ln |z2−1|+i arg(z2−1)).
Since z2 − 1 6= 0 on Ω, ln |z2 − 1| is continuous on Ω. Next we will consider arg(z2 − 1) as the composition
of w = f(z) = z2 − 1 on Ω with g(w) = argw on f(Ω), where the argument is to be chosen later. Now f is
clearly continuous on Ω and f(Ω) = C \[0,+∞). Next we will choose 0 < argw < 2pi on f(Ω) so as to make
20
g continuous on f(Ω). Therefore, arg(z2 − 1) = g (f(z)) is continuous on Ω, log(z2 − 1) is continuous on Ω
and
√
z2 − 1 is continuous on Ω.
Exercises
1. Find all z = x+ iy such that ez = 1. Use this to find all the roots of cos z and sin z.
2. Prove that sin 2z = 2 sin z cos z for all complex number z.
3. Discuss if it is possible to define log(z − 1) continuously on C\[−1, 1]. Also, discuss the possibility for
log
(
z + 1
z − 1
)
to be continuously defined on C \[−1, 1].
4. Let a1, a2, a3, . . . be a bounded sequence in C . For every ε > 0, prove that
∞∑
n=1
ane
−z ln n converges
uniformly on {z : Re z ≥ 1 + ε}.
5. Define cosh z = (ez + e−z)/2, sinh z = (ez − e−z)/2 and tanh z = (sinh z)/(cosh z). Prove that
cosh(z + w) = cosh z coshw + sinh z sinhw and sinh(z +w) = sinh z coshw + cosh z sinhw.
Prove that if z = x+ iy, then | cos z|2 = cos2 x+ sinh2 y and | sin z|2 = sin2 x+ sinh2 y.
6. Find all the roots and periods of cosh z and sinh z.
21
8. Conformal Mappings
Definitions. (1) f is conformal at z0 ∈ C iff f is holomorphic in a neighborhood of z0 and f ′(z0) 6= 0.
(2) For regions U and V, f : U → V is a conformal mapping from U to V iff f is bijective and f is conformal
at each point of U. In that case, we also say U and V are conformally equivalent.
Observation. If f is conformal at z0, then f is angle preserving (in direction and magnitude).
γ
γ
γ
2
( t )
( t )
1( t )0
θ
z
For γ(t) = (x(t), y(t)), γ′(t) = (x′(t), y′(t)). In the figure, we have
tan θ =
dy
dx
=
y′(t)
x′(t)
. So, θ = tan−1
y′(t)
x′(t)
= arg γ′(t).
Suppose γ1(t0) = γ2(t0) = z0. Then the angle between γ1, γ2 at z0 is given by
arg γ′2(t0)− arg γ′1(t0).
2
2
w=f
z
z
γ
γθ γ
γ
0
1 01
( z )
θf
f
So, angle between f ◦ γ1, f ◦ γ2 at w0 = f(z0)
= arg(f ◦ γ2)′(t0) − arg(f ◦ γ1)′(t0)
= arg(f ′(z0)γ′2(t0)) − arg(f ′(z0)γ′1(t0))
= arg f ′(z0) + arg γ′2(t0)− arg f ′(z0) − arg γ′1(t0)
= angle between γ1, γ2 at z0.
The observation has an interesting interpretation: if you lives at z0 and f is a typhoon that blows your
home from z0 to f(z0), then you would not know that you have been moved because all the streets near your
home intersect at the same angles as before!
Examples. It will be convenient to treat lines as circles of infinite radius and a “circle” will mean either a
line or a circle.
(1) w = az + b, a 6= 0 is a conformal mapping from C to C. Writing a = Reiα, b = u + iv, we see that
az + b = R(eiαz) + u+ iv rotates z (with respect to the origin) by angle α, then expands (or contracts)
by a factor of R, then translate by (u, v). This mapping takes a line to a line and a circle to a circle,
i.e. takes “circles” to “circles”.
(2) w =
1
z
is a conformal mapping from C \{0} to C \{0}. The equation α(x2+ y2) + βx+ γy + δ = 0 (line
if α = 0, circle if α 6= 0) is equivalent to αzz+Dz+Dz+ δ = 0, where D = β − iγ
2
. Under the mapping
it becomes δww +Dw +Dw + α = 0. So the map w =
1
z
sends “circles” to “circles”.
z
.
θ
w=az+b
z θ-
w
r
r
1
zw=
1
w
θ
22
(3) w = zα, α > 0 is conformal at all z 6= 0.
ϕ
θ
z
r r α
w
αϕ
αθ
w=zα
(4) w = Log z is conformal at z ∈ C \(−∞, 0].
θ
R
Logw=
pii
θi
z
LogR
z w
(5) Mo¨bius transformations (or linear fractional transformations or bilinear transformations) are functions
of the form w = T (z) =
az + b
cz + d
, where a, b, c, d ∈ C and ad− bc 6= 0.
Properties of Mo¨bius Transformations.
(1) (Basic Property) T (z) =
az + b
cz + d
(ad − bc 6= 0) is a bijective map from C∪{∞} onto C∪{∞}, where
T (∞) =
{ a
c
if c 6= 0
∞ if c = 0
, ∞ =
{
T (
−d
c
) if c 6= 0
T (∞) if c = 0
; T−1(z) =
dz − b
−cz + a, T (T
−1(z)) = z = T−1(T (z));
T ′(z) =
ad− bc
(cz + d)2
6= 0 for z 6= −d
c
. So for c = 0, T (z) is a conformal mapping from C onto C and for
c 6= 0, T (z) is a conformal mapping from C \
{
−d
c
}
onto C\
{a
c
}
.
(2) (Algebraic Property) If T1(z) =
a1z + b1
c1z + d1
(a1d1 − b1c1 6= 0) and T2(z) = a2z + b2
c2z + d2
(a2d2 − b2c2 6= 0),
then T1 ◦ T2(x) = T1(T2(z)) = (a1a2 + b1c2)z + (a1b2 + b1d2)(c1a2 + d1c2)z + (c1b2 + d1d2) .
(Observe that
(
a1 b1
c1 d1
)(
a2 b2
c2 d2
)
=
(
a1a2 + b1c2 a1b2 + b1d2
c1a2 + d1c2 c1b2 + d1d2
)
. This shows that there is a group
homomorphism from the Mo¨bius transformations to the two-by-two matrices.)
(3) (Geometric Property) T (z) =
az + b
cz + d
=

−
(
ad− bc
c
)
1
cz + d
+
a
c
if c 6= 0
a
d
z +
b
d
if c = 0
is a composition of map-
pings of the form Az+B and
1
z
. From examples (1) and (2), we conclude that Mo¨bius transformations
take “circles” to “circles”.
(4) (Analytic Property) A fixed point of T (z) is a point z0 such that T (z0) = z0. The only Mo¨bius trans-
formation having more than two fixed points is the identity mapping.
(Proof. T (z) =
az + b
cz + d
= z ⇐⇒ cz2 + (d − a)z − b = 0, which has at most two roots, unless c = 0,
d− a = 0, b = 0.)
Hence, if two Mo¨bius transformations take the same values on three points of C∪{∞}, then they must
be identical.
23
(Proof. S(z1) = T (z1), S(z2) = T (z2), S(z3) = T (z3) ⇒ S ◦ T−1 has three fixed points T (z1), T (z2),
T (z3)⇒ S ◦ T−1 is the identity mapping ⇐⇒ S(z) ≡ T (z).)
Definitions. For distinct z2, z3, z4 ∈ C∪{∞}, let Sz2,z3,z4(z) =
z − z3
z − z4
/
z2 − z3
z2 − z4 . (Then Sz2,z3,z4 is a Mo¨bius
transformation that takes the “circle” through z2, z3, z4 to the real axis because Sz2,z3,z4 sends z2, z3, z4 to
1, 0,∞, respectively.)
The cross ratio of z1, z2, z3, z4 is (z1, z2, z3, z4) = Sz2,z3,z4(z1) =
z1 − z3
z1 − z4 ×
z2 − z4
z2 − z3 . (If one of z1, z2, z3, z4
is ∞, then we take limit to get the cross ratio.)
Properties of Cross Ratio.
(1) (z, 1, 0,∞) = lim
w→∞(z, 1, 0, w) = limw→∞
z(1 −w)
(z − w) = z.
(2) (z1, z2, z3, z4) = (z1, z2, z3, z4).
(3) (Tz1, T z2, T z3, T z4) = (z1, z2, z3, z4) for any Mo¨bius transformation T .
(Proof. Sz2,z3,z4 ◦ T−1:
Tz2 → 1
Tz3 → 0
Tz4 →∞
. So by the holomorphic property, Sz2,z3,z4 ◦ T−1 = STz2,Tz3,Tz4 . Hence,
evaluating both sides at T (z1), we get Sz2,z3,z4(z1) = STz2,Tz3,Tz4(Tz1).)
(4) z1, z2, z3, z4 are distinct points on a “circle” if and only if (z1, z2, z3, z4) is real. (This is because Sz2,z3,z4
takes the “circle” through z2, z3, z4 to the “circle” through 1, 0,∞, which is the real axis.)
Below we shall show that cross ratio is a useful device that allows us to transform geometrical informa-
tions (such as, symmetry of points and interior/exterior of circles) into algebraic quantities for computation
and derivation of important principles.
c
R
b
w w*
Γ
Definition. For a given circle Γ: |z − c| = R, points w, w∗ ∈ C∪{∞}
are symmetric with respect to Γ iff (w∗−c)(w−c) = R2. This is denoted
by w ∼Γ w∗. Equivalently,w ∼Γ w∗ ⇐⇒ w∗ = R
2
w − c + c. (Observe
that c, w,w∗ are collinear because w∗ − c = R
2
|w− c|2 (w − c). Also,
|w − c|
R
=
R
|w∗ − c| implies ∠cbw
∗ = 90◦.) In the case Γ is a line, w∗ is
taken to be the mirror image of w with respect to Γ.
Properties of Symmetric Points.
(1) The line ww∗ is orthogonal to Γ (because c, w,w∗ are collinear).
(2) A point is the center of Γ if and only if it is symmetric to ∞ with respect to Γ, i.e. w = c ⇐⇒ w∗ =∞.
(3) w ∼Γ w∗ ⇐⇒ for any distinct z2, z3, z4 on Γ, (w∗, z2, z3, z4) = (w, z2, z3, z4).
(Proof. Consider the Mo¨bius transformation T (z) =
R2
z − c + c, then T (
R2
w − c + c) = w. Since z2 on Γ
implies (z2 − c)(z2 − c) = R2, it follows that T (z2) = z2. Similarly, T (z3) = z3 and T (z4) = z4.
(⇒) If w ∼Γ w∗, then (w∗, z2, z3, z4) = (T (w∗), T (z2), T (z3), T (z4)) = (w, z2, z3, z4).
(⇐) We have
(w∗, z2, z3, z4) = (w, z2, z3, z4) = (T (
R2
w − c + c), T (z2), T (z3), T (z4)) = (
R2
w − c + c, z2, z3, z4).
24
Since Sz2,z3,z4 is injective, the definition of cross ratio implies w∗ =
R2
w − c + c, i.e. w ∼
Γ w∗.)
Mo¨bius transformations send pairs of symmetric points to pairs of symmetric points as the following
shows.
Symmetry Principle. w ∼Γ w∗ ⇐⇒ T (w) ∼T (Γ) T (w∗) for every Mo¨bius transformation T .
Proof. By property (3) of symmetric points and cross-ratio, w ∼Γ w∗ iff (w∗, z2, z3, z4) = (w, z2, z3, z4) iff
(T (w∗), T (z2), T (z3), T (z4)) = (T (w), T (z2), T (z3), T (z4)) iff T (w) ∼T (Γ) T (w∗). QED
Mo¨bius transformations need not send the interior of “circles” to the interior of the image “circles”
(e.g. consider w =
1
z
inside the unit circle). However, there is an orientation principle which says Mo¨bius
transformations do send the right sides of oriented “circles” to the right sides of their image “circles”.
Definition. An orientation of a circle Γ is an ordered triple points (z2, z3, z4) on Γ.
By property (4) of cross ratio, Γ = {z: Im(z, z2, z3, z4) = 0}. Now, the right side of Γ with respect to
(z2, z3, z4) (i.e. the side the right hand “touches” as one “walks” on Γ from z2 to z3 to z4) is given by the
set Rz2,z3,z4Γ = {z: Im(z, z2, z3, z4) > 0}.
To see this, take the case (z2, z3, z4) is counterclockwise, then the right side is the outside of Γ, which
includes ∞. Since Sz2,z3,z4(z) = (z, z2, z3, z4) is bijective and sends Γ to the real axis, it must send the
outside of Γ onto either the upper or the lower half plane because of connectivity. (To be precise, suppose
p1, p2 are outside Γ such that Sz2,z3,z4 sends them to q1, q2, one on the upper half plane and the other on
the lower half plane. Take a (continuous) curve joining p1 and p2 without intersecting Γ, then the image of
the curve under Sz2,z3,z4 will not intersect the real axis. However, all curves joining q1 and q2 must intersect
the real axis, which yields a contradiction.)
By definition, Rz2,z3,z4Γ is the side of Γ that is sent to the upper half plane. We intend to show, for the
counterclockwise case, that this is the outside of Γ. For this, it suffices to see ∞ ∈ Rz2,z3,z4Γ.
z 2
z 3
z 4
Γ
o
θ
o
Now observe that (∞, z2, z3, z4) = z2 − z4
z2 − z3 =
z4 − z2
z3 − z2 = Re
iθ, 0 < θ < pi. So,
Im(∞, z2, z3, z4) > 0, which implies ∞ ∈ Rz2,z3,z4Γ.
The clockwise case is similar. Also, the case Γ is a line can be checked, e.g.
R1,0,∞R = {z: Im z > 0} is the upper half plane)
Similarly, the left side of Γ with respect to (z2, z3, z4) is given by Lz2,z3,z4Γ = {z: Im(z, z2, z3, z4) < 0}.
Orientation Principle. For an orientation (z2, z3, z4) on a circle Γ and T a Mo¨bius transformation, we
have T (Rz2,z3,z4Γ) = RT (z2),T (z3),T (z4)T (Γ) and T (Lz2,z3,z4Γ) = LT (z2),T (z3),T (z4)T (Γ). In particular, T must
send each side of Γ onto a side of TΓ.
Proof. By the definition of right side and property (3) of cross-ratio, z ∈ Rz2,z3,z4Γ iff Im(z, z2, z3, z4) =
Im(Tz, T z2, T z3, T z4) > 0 iff Tz ∈ RTz2,Tz3,Tz4T (Γ). By similar reason, we also have z ∈ Lz2,z3,z4Γ iff
Tz ∈ LTz2,Tz3,Tz4T (Γ). QED
Theorem. For real θ, |a| < 1, T (z) = eiθ
(
z − a
1− az
)
is a conformal map of the open unit disk D onto itself.
(Later we will show these are the only conformal maps from D onto D.)
Proof. Observe that T takes the unit circle ∂D to itself because for |z| = 1, |T (z)| =
∣∣∣∣eiθ z − a1− az
∣∣∣∣ =
|z − a|
|zz − az| =
|z − a|
|z − a||z| = 1. Since T (a) = 0, the orientation principle implies T maps D onto D. QED
25
In an advanced course, a theorem called the Riemann Mapping Theorem is usually proved, which asserts
that for every simply connected region Ω 6= C (i.e. Ω has no “holes” in its interior), there is a conformal
mapping f from the open unit disk D onto Ω. That is, every proper simply connected region is conformally
equivalent to D. The mapping between D and Ω is called a Riemann mapping.
Remarks. By composing with another conformal mapping from D onto D, we see that Riemann mappings
are not unique. In fact, if there is one Riemann mapping, there are infinitely many. However, if the values
f(0) ∈ Ω and f ′(0) > 0 are prescribed, then there is a second half of the Riemann mapping theorem, which
asserts that there is exactly one Riemann mapping satisfying these additional conditions.
Later on we will prove many theorems about the open unit disk or functions on it. Because of the
Riemann mapping theorem, they then can be viewed as theorems about arbitrary proper simply connected
regions or functions on them. So, the open unit disk is a “canonical object” in complex analysis.
Let us compute some conformal mappings between D and some regions. Examples (1),(2),(3),(4) and
(6) are Riemmann mappings between the open unit disk D and some common regions. Example (8) shows
an interesting application of the symmetry principle and the orientation principle in conformal mappings.
Examples.
10
o1D-1
Ti
Upper Half
UHP
o
Plane
(1) T (z) = Si,1,−1(z) = i
(
1− z
1 + z
)
,
T−1(z) =
i− z
i+ z
.
D
0
RHP
Right Half
S -1
S
Plane
(2) S(z) =
1
i
T (z) =
1− z
1 + z
,
S−1(z) =
1− z
1 + z
.
D
R
\ (-oC o,0]
(3) R(z) = (S(z))2 =
(
1− z
1 + z
)2
.
D
Q
1 st Quadrant
(4) Q(z) =
√
T (z) =
√
i
(
1− z
1 + z
)
.
(the principal square root)
1-1 0
i
0
i
P
w
z
1
(5) P (z) = i
(
1− z
1 + z
)
, P :
i 7→ 1
1 7→ 0
0 7→ i
−1 7→ ∞
,
P−1(w) =
i −w
i +w
. (c.f. Example (1))
0-1 1
i
P
T -1
D
0
z z²
ϑ
(6) ϑ(z) = T−1(P (z)2) = T−1
(
−
(
1− z
1 + z
)2)
=
−
(
1−z
1+z
)2
− i(
1−z
1+z
)2
− i
= −i z
2 + 2iz + 1
z2 − 2iz + 1
26
NP( z )²
1-1
i
-i
1-1
z -z
1
z
z
(7) N (z) = P
(
−1
z
)2
= −
(
1− (−1z )
1 + (−1z )
)2
= − (z + 1)
2
(z − 1)2 .
. . . .
w*
-6
Γ1
Γ2
-1 w
(8) Let U be the interior of the annulus bounded by Γ1: |z+1| = 9,Γ2: |z+6| = 16.
Find a conformal map from U onto a concentric annulus V centred at the origin.
Solution. The key is first we find a pair of points w and w∗ symmetric with respect to
Γ1 and Γ2. (Such a pair must be collinear with the centers, so they must be real.)
For such a pair, (w+1)(w∗+1) = 81, (w+6)(w∗+6) = 256. Solving we get w = 2, w∗ = 26 (or reverse).
Consider T (z) =
z − 2
z − 26. This is a Mo¨bius transformation, hence takes circles to circles and right sides to
right sides.
Since w = 2
Γ1,Γ2∼ w∗ = 26, the symmetry principle implies T (w) = 0 TΓ1,TΓ2∼ T (w∗) = ∞. By property
(2) of symmetric points, T (w) = 0 is the center of TΓ1, TΓ2. Next, take 8 ∈ Γ1 and 10 ∈ Γ2. Since T (8) = −13
and T (10) = −1
2
, we get TΓ1 = C(0,
1
3
) and TΓ2 = C(0,
1
2
). By the orientation principle, as 8 is inside Γ2
and −1
3
is inside TΓ2, T sends the inside of Γ2 to the inside of TΓ2. As 10 is outside Γ1 and −12 is outside
of TΓ1, T sends the outside of Γ1 to the outside of TΓ1. So, T (U ) = V =
{
w:
1
3
< |w| < 1
2
}
.
Exercises
1. Find conformal mappings from the open unit disk D = {z : |z| < 1} onto the following regions:
(a) the infinite strip {z : 0 < Im z < 1};
i
(b) the upper semidisk {z: |z| < 1 and Im z > 0}.
1-1 0
i
[Hint: Find a map from 1st quadrant onto the upper half semidisk.]
(c) the slit disk D \ [0, 1) 0 1
(d) the finite-slit plane C∪
{∞} \ [−1, 1]
-1 1
[Hint: Find a map from C \(−∞, 0] to (C∪{∞} \ [−1, 1].]
27
2. What is the range of ez if we take z to lie in the infinite strip | Im z| < pi
2
? What are the images of
the horizontal lines and vertical segments in | Im z| < pi
2
under the ez mapping? Give an example of a
holomorphic function from the open unit disk D onto C \{0} with derivative never equal to zero.
3. Find r such that there is a bijective conformal mapping from {z : Im z < 0 and |z + 2i| > 1} onto
{w : r < |w| < 1}.
4. Let a < b and T (z) =
z − ia
z − ib . Define L1 = {z: Im z = b}, L2 = {z: Im z = a}, L3 = {z: Re z = 0}.
Determine which of the regions A, B, C, D, E, F in Figure 1, are mapped by T onto the regions U , V ,
W , X, Y , Z in Figure 2.
0 1
Figure 1 Figure 2
DA
B
C
E
F
L
L
L 3
2
1
U W
X Z
V
Y
T
ia
ib
[Hint: Orient L3 by (∞, ia, ib). ]
28
9. Contour Integrals
Definition. If γ: [a, b]→ C is continuous and is given by γ(t) = u(t) + iv(t), then we define∫ b
a
γ(t) dt =
∫ b
a
u(t) dt+ i
∫ b
a
v(t) dt.
Example. For γ(t) = eit = cos t+ i sin t, 0 ≤ t ≤ pi
2
,
∫ pi
2
0
γ(t) dt =
∫ pi
2
0
cos t dt+ i
∫ pi
2
0
sin t dt = 1 + i.
Triangle Inequality.
∣∣∣∣∣
∫ b
a
γ(t) dt
∣∣∣∣∣ ≤
∫ b
a
|γ(t)| dt.
Proof. Taking real part, Re
∫ b
a
γ(t) dt =
∫ b
a
u(t) dt =
∫ b
a
Reγ(t) dt and similarly for imaginary part. Next,
let I =
∫ b
a
γ(t) dt = |I|eiθ for some real θ. Then triangle equality follows from∣∣∣∣∣
∫ b
a
γ(t) dt
∣∣∣∣∣ = |I| = Re(Ie−iθ) =
∫ b
a
Re(γ(t)e−iθ) dt ≤
∫ b
a
|γ(t)| dt.
Definitions. A continuous curve z: [a, b]→ C, z(t) = x(t) + iy(t), is piecewise smooth iff except for finitely
many points, x′(t) and y′(t) are continuous on [a, b] and z′(t) = x′(t)+ iy′(t) 6= 0 on [a, b]; it is smooth iff no
exception occurs. The curve is closed iff z(a) = z(b). A closed curve is simple iff the function is injective on
[a, b), i.e. it has no self-intersection.
Henceforth all curves will be assumed piecewise smooth and have finite lengths.
Definitions. If C is a smooth curve given by z: [a, b] → C and f is continuous on the curve C, then we
define
∫
C
f(z) dz =
∫ b
a
f(z(t))z′(t) dt and
∫
C
|f(z)| |dz| =
∫ b
a
|f(z(t))| |z′(t)| dt.
Examples. (1) Let C be the unit circle given by z(t) = eit(0 ≤ t ≤ 2pi). Now f(z) = 1
z
is continuous on C.
So
∫
C
1
z
dz =
∫ 2pi
0
1
eit
(ieit) dt = 2pii and
∫
C
∣∣∣∣1z
∣∣∣∣ |dz| = ∫ 2pi
0
dt = 2pi.
(2) Let C be the curve z(t) = 1 + it2 (0 ≤ t ≤ 1) and f(z) = z2. Then
∫
C
f(z) dz =
∫ 1
0
(1 + it2)2(2it dt) =∫ 1
0
(2it − 4t3 − 8it5) dt = (it2 − t4 − 4i
3
t6)
∣∣∣∣1
0
= −1− i
3
.
Properties of Contour Integrals.
(1) If C1, C2 given by z1: [a, b]→ C, z2: [c, d] → C are smoothly equivalent (in the sense that z2 = z1 ◦ λ,
where λ: [c, d]→ [a, b] is injective with continuous derivative and λ(c) = a, λ(d) = b), then by the change
of variable s = λ(t), we have z′2(t) dt = z
′
1(λ(t))λ
′(t) dt = z′1(s) ds. Hence∫
C2
f(z) dz =
∫ t=d
t=c
f(z2(t))z′2(t) dt =
∫ s=b
s=a
f(z1(s))z′1(s) ds =
∫
C1
f(z) dz.
29
(2)
∫
−C
f(z) dz = −
∫
C
f(w) dw, where C, −C are given by w, z: [a, b]→ C and z(t) = w(a+ b− t).
(3)
∫
C
[f(z) + g(z)] dz =
∫
C
f(z) dz +
∫
C
g(z) dz,
∫
C
αf(z) dz = α
∫
C
f(z) dz.
(4) Fundamental Theorem of Calculus.
∫
C
f ′(z) dz=f(z(b)) − f(z(a)), where C is given by a smooth
function z: [a, b]→ C and f is differentiable on C.
Closed Curve Theorem. If C is a closed curve and f has an antiderivative (say F (z)) in a region
containing C, then
∫
C
f(z) dz =
∫
C
F ′(z) dz = F (z(b)) − F (z(a)) = 0. (In particular,
∫
C1
f(z) dz =∫
C2
f(z) dz for curves C1, C2 that have the same initial and terminal points because C1 followed by −C2
is a closed curve.)
Corollary. If C is a closed curve, then
∫
C
P (z) dz = 0 for every polynomial P (z).
(5) M-L Inequality. If |f(z)| ≤M for every z on C and C has length L, then∣∣∣∣∫
C
f(z) dz
∣∣∣∣ =
∣∣∣∣∣
∫ b
a
f(z(t))z′(t) dt
∣∣∣∣∣ ≤
∫ b
a
|f(z(t))||z′(t)| dt =
∫
C
|f(z)||dz| ≤M
∫ b
a
|z′(t)| dt = ML.
(6) If fn(z) converges uniformly to f(z) on C, then lim
n→∞
∫
C
fn(z) dz =
∫
C
f(z) dz.
(Proof. Let L be the length of C. For any ε > 0, because fn(z) converges uniformly to f(z) on C, there
is N such that n ≥ N ⇒ |fn(z) − f(z)| < ε for all z on C. Then for n ≥ N , by the M -L inequality,∣∣∣∣∫
C
fn(z) dz −
∫
C
f(z) dz
∣∣∣∣ = ∣∣∣∣∫
C
(fn(z)− f(z)) dz
∣∣∣∣ ≤ εL.)
Exercises
1. Find
∫
C
1
z2
dz, where C is a smooth curve from 1 to −1 not passing through the origin.
2. Suppose f(z) is holomorphic and |f(z) − 1| < 1 in a region G. Show that
∫
C
f ′(z)
f(z)
dz = 0 for every
closed curve C in G, assuming f ′ is continuous.
3. Evaluate
∫
C
z
z
dz, where C is the simple closed curve that goes from −2 to −1 along the real axis, then
goes in the clockwise direction on the unit circle to 1, then goes from 1 to 2 along the real axis and
finally goes back to −2 in the counterclockwise direction on the circle |z| = 2. See figure below.
[Hint:
∫
C
=
∫
C1
+
∫
C2
+
∫
C3
+
∫
C4
]
i
C4
C1
2C
C3
21-1-2 0
2i
30
4. Find
∫
|z|=1
|z − 1| |dz|, where the unit circle |z| = 1 is given the counterclockwise orientation.
5. If Rea ≤ 0 and Re b ≤ 0, show that |ea − eb| ≤ |a− b|.
6. Suppose f is holomorphic on B(0, r) and Ref ′(z) > 0 for all z ∈ B(0, r). Show that f is injective.
[Hint: Express f(a) − f(b) as an integral.]
7. If f is a continuous real-valued function and |f(z)| ≤ 1, show that
∣∣∣∣∣
∫
|z|=1
f(z) dz
∣∣∣∣∣ ≤
∫ 2pi
0
| sin t| dt = 4.
[Hint: Let I =
∫
|z|=1
f(z) dz, then for some θ, |I| = Ie−iθ = Re(Ie−iθ).]
31
10. Cauchy Theory
Augustine Cauchy developed a body of facts concerning the integration of holomorphic functions and
many useful applications. First, we present the local Cauchy theory. The word “local” refers to the situation
in a neighborhood of a point. We begin with a fundamental property that leads to the whole development.
Definition. (Nonstandard Terminology!) A continuous function f has the rectangle property in a region Ω
iff
∫
Γ
f(z) dz = 0 for every rectangle Γ such that Γ and its inside are contained in Ω and its edges are parallel
to the real or imaginary axes. Such a rectangle will be referred to as a “rectangle” in Ω.
Rectangle Theorem (or Goursat’s Theorem). If f is holomorphic on a region Ω, then f has the
rectangle property in Ω.
Integral Theorem. If f has the rectangle property in a disk B(c,R) (0 < R ≤ ∞), then f has an
antiderivative there.
Cauchy’s Theorem on Disks. If f is holomorphic on a disk B(c,R), then
∫
C
f(z) dz = 0 for every closed
curve C in B(c,R) (by the rectangle, integral and closed curve theorems.)
sΩ
Γ
Proof of the Rectangle Theorem. Let Γ be a “rectangle” in Ω. Let s
be the length of the longer side of Γ. Suppose
∫
Γ
f(z) dz = I. Divide the
interior of Γ into four congruent subrectangular regions with boundaries Γ1,
Γ2, Γ3, Γ4. Since
∫
Γ
f(z) dz =
4∑
j=1
∫
Γj
f(z) dz, one of the integrals must satisfy∣∣∣∣∣
∫
Γj
f(z) dz
∣∣∣∣∣ ≥ |I|4 . Let us define Γ(1) = Γj .
Repeat this subdivision to the interior of Γ(1) to obtain a Γ(2) and so on. We obtain a sequence of
“rectangles” Γ(1),Γ(2),Γ(3), . . . in Ω such that
∣∣∣∣∫
Γ(k)
f(z) dz
∣∣∣∣ ≥ |I|4k . The rectangular regions bounded by the
Γ(k)’s “shrink” to a point z0 in Ω.
Since f is holomorphic at z0, so g(z) =
f(z) − f(z0)
z − z0 − f
′(z0)→ 0 as z → z0. So given ε > 0, there
is δ > 0 such that |z − z0| ≤ δ ⇒ |g(z) − 0| ≤ ε. Observe that the sides of Γ(k) are at most s/2k, so
|z − z0| ≤
√
2s/2k for all z in Γ(k). Choose k large so that
√
2s/2k < δ, then
|I|
4k
≤
∣∣∣∣∫
Γ(k)
f(z) dz
∣∣∣∣ =
∣∣∣∣∣∣∣
∫
Γ(k)
(f(z0) + f ′(z0)(z − z0)︸ ︷︷ ︸
polynomial
+g(z)(z − z0)) dz
∣∣∣∣∣∣∣ =
∣∣∣∣∫
Γ(k)
g(z)(z − z0) dz
∣∣∣∣.
≤
(
ε
√
2s
2k
)(
4
s
2k
)
=
4
√
2s2ε
4k
⇒ |I| ≤ 4√2s2ε.
Since ε is arbitrary,
∫
Γ
f(z) dz = I = 0. QED
32
ab
Re b + i Im a
Proof of the Integral Theorem. Without loss of generality, let c = 0. For w
in the disk, define F (w) =
∫
C0,w
f(z) dz, where Ca,b denotes the curve from a to
Re b+ i Im a, then to b. By the rectangle property,
∫
Γ
f(z) dz = 0 for the boundary
Γ of every rectangle inside the disk.
We will show F ′ = f . Fix w in the disk. For any ε > 0, since f is continuous at w, there is δ > 0 such that
|z−w| ≤ δ ⇒ |f(z)−f(w)| ≤ ε. For |h| ≤ δ, the length of Cw,w+h is at most
√
2|h|. Let v = Re(w+h)+i Imw.
Now
∫
Cw,w+h
f(w) dz =
∫
Cw,v
f(w) dz +
∫
Cv,w+h
f(w) dz = (v −w)f(w) + (w + h − v)f(w) = hf(w).
y
w+h
x
wv
Also, let x = Rew and y = Re(w + h). By the rectangle property, the integral of
f(z) on the path from w to x to y to v is the same as on Cw,v. So,∣∣∣∣F (w + h) − F (w)h − f(w)
∣∣∣∣ =
∣∣∣∣∣1h
∫
Cw,w+h
(f(z) − f(w)) dz
∣∣∣∣∣ ≤ 1|h|ε · √2|h| = √2ε.
Therefore, F ′(w) = lim
h→0
F (w + h) − F (w)
h
= f(w). QED
Occasionally, we come across some continuous functions that are “almost” holomorphic in their domains
(with exceptions only on some “thin” subsets), it is remarkable that such functions still have the rectangle
property on their domains, as the following theorem asserts.
Extension Theorem. If f is continuous on a region D and holomorphic on D \L, where L is a (horizontal)
line segment (or in the extreme case, a point), then f has the rectangle property on D.
upper
lower
L
L
L
Γ
Γ
Γ
Γ
e ε
e
y 0
x 0 x 1
Proof. Let Γ be a “rectangle” in D. There are three cases to consider.
(1) If Γ and its interior do not contain any point of L, then Γ is a “rectangle”
in D \ L and
∫
Γ
f(z) dz = 0 by the rectangle theorem.
(2) If Γ has a (horizontal) edge e containing points of L, let Γε be the rectangle
that has the same edges as Γ except the edge e is moved to an edge eε, which
is ε unit from L on the same side as Γ. By (1),
∫
Γε
f(z) dz = 0. Now as ε→ 0,
the (uniform) continuity of f on e implies∫
eε
f(z) dz =
∫ x0
x1
f(x + i(y0 + ε)) dx→
∫ x0
x1
f(x + iy0) dx =
∫
e
f(z) dz.
Since 0 =
∫
Γε
f(z) dz →
∫
Γ
f(z) dz as ε→ 0, we get
∫
Γ
f(z) dz = 0.
(3) If the interior of the vertical edges of Γ contain points of L, then
∫
Γ
f(z) dz =∫
Γupper
f(z) dz +
∫
Γlower
f(z) dz = 0, where we used L to cut Γ into “rectangles”
Γupper and Γlower having horizontal edges on L and applied (2). QED
Now we turn to the (global) Cauchy theory, which deals with the integration of holomorphic functions
on (closed) curves in (simply connected) domains. The curves need not be restricted to lie inside disks.
Definition. A continuous function f defined on a region D has the polygon property iff
∫
Γ
f(z) dz = 0 for
every right-angled polygon Γ such that Γ and its inside are contained in D.
33
Polygon Theorem. Every holomorphic function on a simply connected region has the polygon property.
Proof. Subdivide the inside of every right-angled polygons into rectangular regions. Then apply the rectangle
theorem. QED
Integral Theorem. Every funcion f having the polygon property on a simply connected region D has an
antiderivative F on D.
Proof. Fix z0 ∈ D. Define F (z) =
∫
C
f(w) dw, where C is a polygonal line from z0 to z consisting of finitely
many horizontal or vertical segments. (If C˜ is another such polygonal line, then C and C˜ form finitely many
right-angled polygons. Hence
∫
C
f(w) dw −
∫
C˜
f(w) dw = 0 and F is well-defined.) That F ′ = f follows by
the same argument as in the first integral theorem. QED
Cauchy’s Theorem. If f is holomorphic on a simply connected region D and C is a closed curve in D,
then (by the polygon, integral and closed curve theorems,)
∫
C
f(z) dz = 0.
b0
b1
Cauchy’s Theorem for Homotopic Curves. If C1, C2 are two curves with the
same initial and terminal points in a simply connected region D, then
∫
C1
f(z) dz =∫
C2
f(z) dz for every holomorphic function f on D. (This follows because C1 and −C2
form a closed curve, so we may apply Cauchy’s theorem above.)
Loop Theorem. Suppose f is holomorphic on C \{a1, . . . , an} and Γ is a simple closed curve surrounding
a1, . . ., an as shown. For each aj, let Cj be a simple closed curve about aj inside Γ.
Γ
a1 a2 anC C
C
...
1 2
n
Then we have ∫
Γ
f(z) dz =
n∑
j=1
∫
Cj
f(z) dz,
where the orientation of Γ, C1, . . ., Cn are as shown.
...
...
a1 a2 an
Proof. Introducing n + 1 cross-cuts as shown on the left and applying
Cauchy’s theorem to the upper and lower simple closed curves, we get
(after cancelling the integrals over the cross-cuts)
0 =
∫
Γ
f(z) dz −
n∑
j=1
∫
Cj
f(z) dz.
The result follows. QED
As an appication of the (global) Cauchy theorems, we will present a theorem that deals with a global
question. For a holomorphic function without roots in a domain, locally (i.e. near every point) we can take
the logarithm of the function. It is of interest to know if there is a continuous logarithm of the function on
the whole domain.
Logarithm Theorem. Let f be holomorphic on a simply connected region D without any root in D. Fix
z0 ∈ D and choose a choice of log f(z0), then G(z) =
∫
C
f ′(w)
f(w)
dw + log f(z0) (where C is any curve in D
from z0 to z) defines a holomorphic branch of log f(z) on D (i.e. eG(z) = f(z) for all z ∈ D). In particular,
if f is entire without any root, then f = eG for some entire function G.
34
Proof. Since
f ′
f
is holomorphic on D, the proof of the integral theorem implies G′ ≡ f
′
f
. Consider the
function A(z) = e−G(z)f(z). We have A′(z) = e−G(z)(−G′(z)f(z)+f ′(z)) ≡ 0. Since A(z0) = e−G(z0)f(z0) =
e− log f(z0)f(z0) = 1 and D is connected, A ≡ 1. Therefore eG ≡ f . QED
Remarks. (1) If D is simply connected and not containing 0, then log z =
∫ z
z0
dw
w
+ log z0 is a holomorphic
logarithm on D. Again because of simply connectedness, the path of integration can be any curve in D from
z0 to z.
(2) For α ∈ C, f(z)α = eα log f(z) wherever log f is defined.
Example. C \[−1, 1] is not a simply connected region. Nonetheless, we will show that there are continuous
(in fact, holomorphic) branches of log
(
z + 1
z − 1
)
and
√
z + 1
z − 1 on C\[−1, 1].
Observe that Ω = C \ ((−∞,−1]∪ [1,+∞)) is simply connected. Now R : C \[−1, 1] → Ω given by
R(z) = 1/z is injective and holomorphic. The function
z + 1
z − 1 on C \[−1, 1] corresponds to
1/z + 1
1/z − 1 =
1 + z
1− z
in Ω, which has no root on Ω as −1 6∈ Ω. By the logarithm theorem, there is a holomorphic branch G(z) =
log
(
1 + z
1− z
)
on Ω. Then F (z) = G(R(z)) = log
(
z + 1
z − 1
)
and e
1
2F (z) are holomorphic on C \[−1, 1].
Exercises
1. Find
∫
C
1
z
dz, where C is a curve that go through a regular n-gon (n ≥ 3) with center at the origin once
in the counterclockwise direction.
2. Find
∫
C
1
z2 − 1 dz, where C is a curve that go around a circle with center at the origin and radius 2
once in the counterclockwise direction.
3. Let f be holomorphic on a simply connected region D without any root in D. For every positive integer
n, prove that there exists a holomorphic function h : D → C such that for every z ∈ D, h(z)n = f(z).
[Hint: Logarithm theorem.]
4. Let f : B(0, 1)→ B(0, 1) be holomorphic. Prove that there exists a holomorphic function g : B(0, 1)→ C
such that for every z ∈ B(0, 1), f(z)3 + g(z)3 = 1. [Hint: Use exercise 3.]
5. Give two continuous branches of
√
z(z − 1) on C \[0, 1]. [Hint: Consider f : C \[0, 1] → C \(−∞, 0]
defined by f(z) = 1− 1/z. Use exercise 3.]
6. Prove that on C \([0, 1]∪ [2,+∞)), there are at least six holomorphic branches of
√
z(z − 1) 3√z − 2.
35
11. Power Series Representation
Below we will show that holomorphic functions have power series representations locally. First we intro-
duce the concept of the winding number of a closed curve and deduce an integral formula for a holomorphic
function. Then the power series represntation follows by applying the formula for summing geometric series
to the integrand in the integral formula.
Definition. Suppose C is a closed curve in C and a 6∈ C. The number n(C, a) = 12pii
∫
C
dz
z − a is called the
winding number of C around a.
Example. Let C: z(t) = a + Reint (0 ≤ t ≤ 2pi). This is the circle of radius R centered at a that “winds”
around a |n| times (counterclockwise if n > 0, clockwise if n < 0.) We have
n(C, a) =
1
2pii
∫
C
dz
z − a =
1
2pii
∫ 2pi
0
1
Reint
Rineint dt = n.
Theorem. For any closed curve C and a 6∈ C, n(C, a) is an integer.
Proof. Let C be given by z: [0, 1]→ C. Since C is closed, z(0) = z(1). For 0 ≤ s ≤ 1, define
F (s) =
∫
C|[0,s]
dz
z − a =
∫ s
0
z′(t)
z(t) − a dt.
Then F (0) = 0 and F ′(s) =
z′(s)
z(s) − a . This implies
d
ds
((z(s)−a)e−F (s)) = (z′(s)−(z(s)−a)F ′(s))e−F (s) ≡ 0.
So (z(s) − a)e−F (s) is constant. Then (z(1) − a)e−F (1) = (z(0) − a)e−F (0) = z(0) − a = z(1) − a 6= 0 as
a 6∈ C. This implies e−F (1) = 1. So F (1) = 2piik, where k ∈ Z . Then n(C, a) = 12piiF (1) = k. QED
a
C
z (0)= z (1)
Remark. Integrating both sides of F ′(s) =
z′(s)
z(s) − a , then using F (0) = 0, we get
F (s) = log
(
z(s) − a
z(0)− a
)
= ln
∣∣∣∣ z(s) − az(0)− a
∣∣∣∣+ i arg(z(s) − az(0) − a
)
.
Now z(s) continuous implies the arg-term is continuous. After we trace z(s)
from s = 0 to 1, arg((z(1)− a)/(z(0)− a)) gives the total change in argument from
z(0) to z(1) around a. Dividing by 2pi, this is the winding number n(C, a).
Cauchy’s Integral Formula. If f is holomorphic on B(c,R), 0 < R ≤ ∞ and C is a closed curve in
B(c,R) \ {a}, then ∫
C
f(z)
z − a dz = 2pii n(C, a)f(a).
Proof. The function g(z) =
{
f(z) − f(a)
z − a if z 6= a
f ′(a) if z = a
is holomorphic on B(c,R) \ {a} and continuous on
B(c,R). By the extension, integral and closed curve theorems,
∫
C
f(z) − f(a)
z − a dz =
∫
C
g(z) dz = 0. There-
fore,
∫
C
f(z)
z − a dz =
∫
C
f(a)
z − a dz = 2pii n(C, a)f(a). QED
36
Power Series Representation. If f is holomorphic on the disk B(a,R), 0 < R ≤ ∞,
then f(w) =
∞∑
n=0
f (n)(a)
n!
(w − a)n (is infinitely differentiable) for all w in B(a,R).
a+ra
w
a+R
Proof. For w in B(a,R), let r be such that |w − a| < r < R. Then the circle |z − a| = r, z(t) = a + reit
winds about a and w once. For z on this circle,
∣∣∣∣w − az − a
∣∣∣∣ = |w − a|r < 1 and 1z −w = 1z − a
(
1
1− w−az−a
)
=
1
z − a
∞∑
n=0
(
w − a
z − a
)n
. By Cauchy’s integral formula,
f(w) =
1
2pii
∫
|z−a|=r
f(z)
z − wdz =
1
2pii
∫
|z−a|=r
∞∑
n=0
(w − a)nf(z)
(z − a)n+1 dz =
∞∑
n=0
(
1
2pii
∫
|z−a|=r
f(z)
(z − a)n+1 dz︸ ︷︷ ︸
βn
)
(w − a)n.
(The last equality follows from uniform convergence for all z on the circle by Weierstrass M-test
∞∑
n=0
∣∣∣∣ (w − a)n(z − a)n+1 f(z)
∣∣∣∣ ≤ ∞∑
n=0
|w− a|n
rn+1
M =
M
r − |w − a| <∞,
where M = max
|z−a|=r
|f(z)|.) The power series representation f(w) =
∞∑
n=0
βn(w − a)n is valid for all w ∈
B(a, r). Therefore, by Taylor’s theorem for power series, βn =
f (n)(a)
n!
. Finally, we let r → R.
QED
Examples of Power Series of Common Functions.
ez =
∞∑
n=0
zn
n!
(all z), cos z =
∞∑
n=0
(−1)nz2n
(2n)!
(all z), sin z =
∞∑
n=0
(−1)nz2n+1
(2n+ 1)!
(all z),
cosh z =
ez + e−z
2
=
∞∑
n=0
z2n
(2n)!
(all z), sinh z =
ez − e−z
2
=
∞∑
n=0
z2n+1
(2n+ 1)!
(all z),
Log z =
∞∑
n=1
(−1)n−1(z − 1)n
n
(for |z − 1| < 1), (1 + z)α = 1 +
∞∑
n=1
α(α− 1) . . . (α− n+ 1)
n!
zn (for any com-
plex α, |z| < 1.)
Corollary 1. Holomorphic functions are infinitely differentiable (because power series are).
This is an amazing fact that is certainly not true for differentiable functions of real variables!
Corollary 2. If f is holomorphic on B(a, r), then g(z) =
{
f(z) − f(a)
z − a if z 6= a
f ′(a) if z = a
=
∞∑
n=0
f (n+1)(a)
n!
(z − a)n
is holomorphic at a (and on B(a, r)).
Corollary 3. If f(z) is holomorphic on Ω and has roots z1, z2, . . . , zk, then there is a holomorphic function
gk(z) on Ω such that f(z) ≡ (z − z1)(z − z2) . . . (z − zk)gk(z).
Proof. For n = 1, g1(z) =
 f(z) − f(z1)z − z1 if z 6= z1
f ′(z1) if z = z1
is holomorphic on Ω. The inductive step follows by
observing that gn(z) = (z − zn+1)gn+1(z) and getting gn+1 as in case n = 1. QED
37
L’ Hoˆpital’s Rule. If f(z) and g(z) are holomorphic in an open set Ω and for some a ∈ Ω, there exists
n ∈ N such that g(n)(a) 6= 0 and f (k)(a) = 0 = g(k)(a) for all k < n, then lim
z→a
f(z)
g(z)
=
f (n)(a)
g(n)(a)
.
Proof. Near a, the power series of f(z) is
∞∑
k=n
f (k)(a)
k!
(z − a)k = f
(n)(a)
n!
(z − a)n + F (z)(z − a)n+1 and the
power series of g(z) has a similar form. Then
lim
z→a
f(z)
g(z)
= lim
z→a
f (n)(a) + n!F (z)(z − a)
g(n)(a) + n!G(z)(z − a) =
f (n)(a)
g(n)(a)
. QED
Definition. If a holomorphic function f(z) has a root a such that f(a) = f ′(a) = . . . = f (n−1)(a) = 0 but
f (n)(a) 6= 0 , it is a root of order (or multiplicity) n. (Equivalently, this means that f(z) = f
(n)(a)
n!
(z − a)n+
f (n+1)(a)
(n+ 1)!
(z − a)n+1 + · · · near a with f (n)(a) 6= 0 or f(z) = (z − a)ng(z) with g(a) 6= 0.) A root of order
one, two or three is called a simple, double or triple root, respectively.
Example. Find lim
z→0
(sin z2)(cos z − 1)
ez4 − 1 .
When w is near 0, sinw = w − w
3
3!
+ . . . , cosw − 1 = −w
2
2!
+
w4
4!
− . . . and ew − 1 = w + w
2
2!
+ . . . . So
lim
z→0
(sin z2)(cos z − 1)
ez4 − 1 = limz→0
(z2 − z
6
3!
+ . . . )(−z
2
2!
+
z4
4!
− . . . )
z4 +
z8
2!
+ . . .
= lim
z→0
−z
4
2
+ . . .
z4 + . . .
= −1
2
.
(Incidentally, from the power series, we can also see that 0 is a double root for sin z2, cos z − 1 and a root of
order 4 for ez
4 − 1.)
Exercises
1. The following curve C divides the plane into 4 regions. For each region, state the winding number of C
around points in that regions. (Give answers by inspection, no computation is needed.)
1 2 3
4
C
2. Find lim
z→0
z Log(z49 + 1)
(cos z25)− 1 .
3. Find
∫
|z|=1
sin z
z
dz,
∫
|z|=1
sin z
z2
dz and
∫
|z|=2
sin z
z2 − 1 dz, where the circles are oriented counterclockwisely.
4. If f is holomorphic on the open unit disk D and |z|+ |f(z)| ≤ 1 for all z ∈ D, then show that f ≡ 0.
5. Suppose f is holomorphic on D = {z : |z| < 1} and f is an even function (i.e. f(z) = f(−z)). Show
that there is a holomorphic function g on D such that g(x) = f(
√
x) for all positive real numbers x < 1.
38
6. Prove that f(z) =
∫ 1
0
sin zt
t
dt is an entire function by obtaining a power series expansion for f .
7. Suppose f is holomorphic on the closed unit disk {z : |z| ≤ 1} and |a| < 1, show that (1− |a|2)|f(a)| ≤
1
2pi
∫ 2pi
0
|f(eiθ)| dθ. [Hint: Consider integrating f(z)1 − az
z − a .]
8. Let f(z) = u(z) + iv(z) (or f(x, y) = (u(x, y), v(x, y))) be a bijec-
tive holomorphic function from the open unit disk D = {z: |z| < 1}
onto a domain G with finite area.
(a) Show that Jf (x, y)
def=
∣∣∣∣∣∣∣
∂u
∂x
∂u
∂y
∂v
∂x
∂v
∂y
∣∣∣∣∣∣∣ = |f ′(z)|2.
y
x
D G
u
vf
(b) For distinct nonnegative integers m, n, show
∫
D
zmzn dA = 0 (orthogonality relation), where dA =
r dr dθ = dx dy is the area differential.
(c) Show that if f(z) =
∞∑
n=0
cnz
n is the power series for f in D, then area of G = pi
∞∑
n=1
n|cn|2.
9. Suppose f is holomorphic on the open unit disk {z : |z| < 1} such that
∞∑
n=0
∣∣∣∣f (n)(0)n!
∣∣∣∣2 <∞. Prove that
lim
r→1−
1
2pi
∫ 2pi
0
|f(reiθ)|2 dθ =
∞∑
n=0
∣∣∣∣f (n)(0)n!
∣∣∣∣2 .
10. Suppose f(z) =
∞∑
n=0
anz
n and g(z) =
∞∑
n=0
bnz
n converges for all z ∈ B(0, 1). Let |z0| < r < 1.
(a) Show that
∞∑
n=0
bnf(w)
zn0
wn+1
converges uniformly on the circle {w : |w| = r}.
(b) Show that
1
2pii
∫
|w|=r
f(w)g
(z0
w
) dw
w
=
∞∑
n=0
anbnz
n
0 .
11. Prove limsup
n→∞
∣∣∣∣∣∣
M∑
j=1
anj
∣∣∣∣∣∣
1/n
= max
j=1,···,M
|aj|. [Hint: Consider the power series with
M∑
j=1
anj as coefficients.]
39
12. Consequences of Cauchy Theory
Cauchy Integral Formula for Derivatives. If f is holomorphic on B(a,R), then for n = 0, 1, 2, . . . and
0 < r < R, we have f (n)(a) =
n!
2pii
∫
|z−a|=r
f(z)
(z − a)n+1 dz (since both sides equal n!βn as shown in the proof of
the power series representation).
Notice this formula said something amazing, namely the derivative at a point can be given by an integral!
We now give an example which illustrates the fact that the Cauchy integral formula for derivatives is
also true for closed curves in a region, provided winding number factors are included. This will be proved
later.
Example. Let f be entire. Find
∫
C
f(z)
(z − a)20 dz, where C is the curve a
b0
.
Solution. Observe that
∫
C
=
∫
C1
+
∫
C2
, where C1 is the curve a
b0
and C2 is the curve a
b0
.
Since
f(z)
(z − a)20 is holomorphic on C\{a}, we can draw a counterclockwise circle C(a, ε) = {z : |z − a| = ε}
winding around a once inside C2. By the loop theorem,
∫
C1
=
∫
|z−a|=ε
=
∫
C2
. So,
∫
C
f(z)
(z − a)20 dz = 2
∫
|z−a|=ε
f(z)
(z − a)20 dz = 2pii n(C, a)
f (19)(a)
19!
.
When combined with the M -L inequality, the Cauchy integral formula for derivatives gives estimates
on the size of derivatives or coefficients of power series.
Liouville’s Theorem. A bounded entire function is constant. (More generally, if f(z) is entire and if there
are constants A and B and α ≥ 0 such that |f(z)| ≤ A+B|z|α when |z| is large (i.e. |z| > r for some r > 0),
then f(z) is a polynomial of degree k ≤ α.)
Proof. Since f(z) is holomorphic on B(0,∞), f(z) =
∞∑
n=0
f (n)(0)
n!
zn for all z (in B(0,∞)). We will show
that f (n)(0) = 0 for n > α. This implies f(z) is a polynomial of degree at most k ≤ α.
Let n > α and R > r. By the Cauchy integral formula for derivatives and M -L inequality, as R→∞,
∣∣∣f (n)(0)∣∣∣ =
∣∣∣∣∣∣∣
n!
2pii
∫
|z|=R
f(z)
zn+1
dz
∣∣∣∣∣∣∣ ≤
n!
2pi
(
A +BRα
Rn+1
2piR
)
=
n!(A+BRα)
Rn
→ 0.
(We can take limit as R → ∞ because f(z) is defined in the whole plane, hence the circle |z| = R would
always be in the domain of f(z), so Cauchy integral formula for derivatives can be applied!) Therefore,
f (n) = 0 and f(z) = f(0) + f ′(0)z + . . .+ f (k)(0)zk with k ≤ α. QED
40
In an advanced course, a theorem called the Little Picard Theorem is usually proved, which asserts that
an entire function whose range omits two complex numbers must be constant. This is a big improvement
over Liouville’s theorem, since bounded functions are functions whose ranges are bounded sets, hence omit
many complex numbers. An example of an entire function whose range omits only one value is ez . Its range
is C \{0}, hence omits only 0.
Fundamental Theorem of Algebra. Every nonconstant polynomial P (z) with complex coefficients has
a root in C.
Proof. Suppose P (z) has no root, then f(z) =
1
P (z)
is entire. Let P (z) = anzn + an−1zn−1+ · · ·+ a0 with
n ≥ 1, an 6= 0. As z →∞, eventually |z| > 1, then
|P (z)| ≥ |anzn| − |an−1zn−1| − · · · − |a0| = |z|n−1
(
|anz| − |an−1| − · · · −
∣∣∣ a0
zn−1
∣∣∣)
≥ |z|n−1(|anz| − |an−1| − · · · − |a0|)→∞.
So, lim
z→∞P (z) =∞ and limz→∞
1
P (z)
= 0. Then |f(z)| ≤ 1 when |z| is large. By Liouville’s theorem, f(z) is a
polynomial of degree 0, i.e. constant. Then P (z) =
1
f(z)
is constant, a contradiction. QED
Identity Theorem. Suppose f is holomorphic on a region D and the roots of f has a limit point in D (i.e.
there are distinct an ∈ D such that f(an) = 0 and an → a ∈ D), then f ≡ 0.
Remark. We have proved the case D = B(a, r) in the identity theorem for power series in chapter 5.
Proof of the Identity Theorem. Let A = {z ∈ D: z is a limit of roots of f(z)} and B = D \ A. Then
A ∪B = D,A ∩B = ∅ and A 6= ∅ by hypothesis.
If a ∈ A, then a = lim
n→∞an with f(an) = 0. By the identity theorem for power series, f(z) = 0 in an
open disk around a. Then every point in this disk belongs to A. So A must be open.
If b ∈ B, then there is a disk B(b, δ) such that f(z) 6= 0 for 0 < |z − b| < δ. Then all points of this disk
cannot be in A. Hence they must be in B. So B must be open.
Since D is connected, B = ∅ and A = D. Therefore f ≡ 0. QED
Uniqueness Theorem. Suppose g and h are holomorphic on a region D and {z: g(z) = h(z)} has a limit
point in D, then g ≡ h (by considering f = g − h).
Examples. (1) The functions g(z) = sin 2z and h(z) = 2 sin z cos z are entire. Since g(x) = h(x) for every
real number x and the real numbers have limit points (e.g.
1
n
→ 0 as n→∞), by the uniqueness theorem,
g ≡ h. Therefore sin 2z = 2 sin z cos z for all complex number z.
(2) The function f(z) = sin (
1
z
) is holomorphic on D = C \{0}. Its roots are 1
npi
(n ∈ Z\{0}). As n →∞,
1
npi
→ 0, but we cannot conclude f(z) ≡ 0 because 0 is not in D.
Mean Value Theorem. If f is holomorphic on a region D containing B(a, r), then (by the Cauchy integral
formula) f(a) =
1
2pii
∫
|z−a|=r
f(z)
z − a dz =
1
2pi
∫ 2pi
0
f(a + reiθ) dθ, i.e. the average value of f on C(a, r) is f at a.
41
Maximum Modulus Theorem. If f is a nonconstant holomorphic function on a region D and B(a, r)
is contained in D, then |f(a)| < |f(w)| for some w in B(a, r). (In addition, if D is bounded and f is also
continuous on the boundary ∂D, then the maximum of |f(z)| can only (and must) occur on ∂D.)
Proof. Let 0 < r′ < r and M = max
0≤θ≤2pi
|f(a + r′eiθ)|. Since |f(a+ r′eiθ)| is a continuous function on [0, 2pi],
M = |f(a + r′eiθ0)|, 0 ≤ θ0 ≤ 2pi. Observe that
|f(a)| = 1
2pi
∣∣∣∣∫ 2pi
0
f(a + r′eiθ) dθ
∣∣∣∣ ≤ 12pi
∫ 2pi
0
|f(a + r′eiθ)| dθ ≤ 1
2pi
∫ 2pi
0
M dθ = M.
If |f(a)| < M , then |f(a)| < |f(w)| for w = a+ r′eiθ0 and we are done. If |f(a)| = M , then |f(a+ r′eiθ)| ≡
|f(a)| because |f(a + r′eiθ)| is continuous in θ. If this is true for all r′ < r, then |f | ≡ |f(a)| on B(a, r)
implies f ≡ f(a) on B(a, r) by the Cauchy-Riemann equations and hence on D by the uniqueness theorem,
contradicting the hypothesis. QED
Minimum Modulus Theorem. If f is a nonconstant holomorphic function on a region D and B(a, r) is
contained in D, then |f(a)| > |f(w)| for some w in B(a, r) unless f(a) = 0. (In addition, if D is bounded
and f is also continuous on the boundary ∂D, then either f has a root in D or the minimum of |f(z)| can
only occur on ∂D.)
Proof. If f(a) 6= 0, then f(z) 6= 0 in some subdisk of B(a, r). Apply the maximum modulus theorem to
1
f(z)
on the subdisk. QED
Open Mapping Theorem. If f is a nonconstant holomorphic function on a region U , then f(U ) is an
open set. (In particular, f(U ) has no boundary points.)
Proof. Let w ∈ U . (If infinitely many of the circles Ck = C(w, 1/k) has a point wk such that f(wk) = f(w),
then since wk → w ∈ U , the uniqueness theorem will imply f(z) ≡ f(w).) Now, since f is nonconstant, so
there must be a circle C = C(w, 1/k) such that f(z) 6= f(w) for all z on C. Let m = 1
2
min
z∈C
|f(z) − f(w)|.
We claim B(f(w),m) ⊆ f(U ). (This will imply all points in f(U ) are not on the boundary of f(U ), hence
f(U ) is open.)
w
D
C
| f ( z ) |- p >_ m Let p ∈ B(f(w),m). Consider the minimum d of |f(z)−p| on B(w, 1/k). For
z ∈ C, |f(z)− p| ≥ |f(z)− f(w)| − |f(w)− p| ≥ 2m−m = m > |f(w)− p|. So the
minimum of |f(z) − p| is not on C. By the minimum modulus theorem, f(z) − p
has a root in B(w, 1/k) ⊆ U . Then p ∈ f(U ). So B(f(w),m) ⊆ f(U ). QED
In particular, the open mapping theorem implies the range of a nonconstant holomorphic function on
a region cannot be an arc or a curve or any closed set. We saw this before by using the Cauchy-Riemann
equations.
Schwarz’s Lemma. Let D be the open unit disk. If f :D → D is holomorphic and f(0) = 0, then |f(z)| ≤ |z|
for all z ∈ D and |f ′(0)| ≤ 1. If |f(z)| = |z| for some z 6= 0 or |f ′(0)| = 1, then f(z) ≡ eiθz for some θ ∈ R .
Proof. Define g(z) =
{
f(z)
z
if 0 < |z| < 1
f ′(0) if z = 0
, then g is holomorphic on D. On the circle |z| = r < 1,
|g(z)| = |f(z)||z| ≤
1
r
. Maximum modulus theorem implies |g(z)| ≤ 1
r
for |z| ≤ r. Let r → 1−, we get
|g(z)| ≤ 1 for |z| < 1, which implies |f(z)| ≤ |z| and |f ′(0)| ≤ 1.
If |f(z)| = |z| for some z ∈ D or |f ′(0)| = 1, then |g(z)| = 1 for some z ∈ D. The maximum modulus
theorem implies g ≡ constant (≡ eiθ), then f(z) ≡ eiθz. QED
42
Earlier we proved that for |a| < 1, θ real, f(z) = eiθ z − a
1− az is a conformal map of the open unit disk D
onto D. Here we will prove the converse.
Theorem. If f is a bijective holomorphic function (i.e. conformal map) from D onto D, then f(z) =
eiθ
z − a
1− az for some θ ∈ R and a ∈ D.
Proof. Let a ∈ D such that f(a) = 0. Let T (z) = z + a
1 + az
, which maps D onto D, the unit circle onto the
unit circle and T (0) = a. The functions g = f ◦ T and g−1 are bijective holomorphic functions of D onto
D such that g(0) = 0 = g−1(0). By Schwarz’s lemma, |g(z)| ≤ |z| and |g−1(z)| ≤ |z| ( ⇐⇒ |z| ≤ |g(z)|).
So the equality |g(z)| = |z| holds for all z ∈ D. Then f ◦ T (z) = g(z) ≡ eiθz. Setting w = T (z), we get
f(w) = eiθT−1(w) = eiθ
w − a
1− az . QED
Remarks. If we recall that every proper simply connected domain is conformally equivalent to the open unit
disk D by the Riemann mapping theorem, then we see that Schwarz’s lemma is not just about holomorphic
mappings from D to D, but it can be viewed as a statement about mappings from simply connected domains
to simply connected domains.
This is also true of many other theorems in complex analysis with settings on D. For theorems that are
true for all simply connected domains, often all we need to prove is the case for the open unit disk, then the
general case usually follows by the Riemann mapping theorem and conformal mappings.
Example. If f is holomorphic on the right half plane such that |f(z)| ≤ 1 and f(1) = 0, what is the
maximum of |f(2)| as all such functions f are considered? Which ones attain the maximum?
T
f
1
Solution. Let T (z) =
1− z
1 + z
, which maps the unit disk D onto the
right half plane and let g(z) = f(T (z)), then g is holomorphic on D,
|g(z)| ≤ 1 and g(0) = 0. By Schwarz’s lemma, |f(T (z))| = |g(z)| ≤ |z|.
Solving T (z) = 2, we get z = −1
3
. Then |f(2)| = |g(−1
3
)| ≤ 1
3
. For
max |f(2)| = 1
3
, we need |g(−1
3
)| = 1
3
= | − 1
3
|. By the equality case
of Schwarz’s lemma, f(T (z)) = g(z) = eiθz for θ ∈ R . Therefore,
f(w) = eiθT−1(w) = eiθ
1− w
1 + w
.
Exercises
1. If the range of an entire function lies in the right half plane Rew > 0, show that the function is a
constant function. [Hint: Compose with a Mo¨bius mapping.]
2. Suppose a polynomial is bounded by 1 on the open unit disk. Show that all of its coefficients are
bounded by 1.
3. Show that if f is holomorphic on {z : |z| ≤ 1}, then there must be some positive integer k such that
f
(
1
k
)
6= 1
k + 1
.
4. Suppose f is holomorphic on the annulus {z : 1 ≤ |z| ≤ 2}, |f(z)| ≤ 1 for |z| = 1 and |f(z)| ≤ 4 for
|z| = 2. Prove that |f(z)| ≤ |z|2 throughout the annulus.
43
5. Let D be the open unit disk. If f : D → D is holomorphic with at least two fixed points (i.e. points w
such that f(w) = w), show that f(z) ≡ z. [Hint: By composing with a suitable Mo¨bius mapping, one
of the fixed points may be moved to the origin.]
6. Let f be an entire function which is real on the real axis and imaginary on the imaginary axis, show
that f is an odd function, i.e. f(z) = −f(−z).
7. Suppose f is a nonconstant holomorphic function on the closed annulus A = {z : 1 ≤ |z| ≤ 2}. If f
sends the boundary circles of A into the unit circle, show that f must have a root in A.
8. Use the uniqueness theorem to prove the identity sin(w + z) = sinw cos z + cosw sin z for all complex
numbers w and z.
9. Find
∫
|z|=1
dz
z4 + 4z2
, where the unit circle is given the counterclockwise orientation.
10. Find
∫ 2pi
0
cos(cos θ) cosh(sin θ) dθ. [Hint: Consider cos eiθ.]
11. Suppose f and g are holomorphic on the closed unit disk. Show that |f(z)|+ |g(z)| takes its maximum
on the boundary. [Hint: Consider f(z)eiα + g(z)eiβ for appropriate α and β.]
12. Suppose f is entire and |f ′(z)| ≤ |z| for all z . Show that f(z) = a+ bz2 with |b| ≤ 1
2
.
13. Find the maximum of
∣∣∣∣f (12
)∣∣∣∣, where f is holomorphic on D = {z : |z| < 2}, f(1) = 0 and |f(z)| ≤ 10
for z ∈ D.
14. Find all holomorphic function(s) f defined on the open unit disk D satisfying f
(
1
2
)
=
2
3
and f(z) =
(2− f(z))f(2z) for all z ∈ D.
15. Let H = {z : Re z > 0}. Suppose f : H → H is holomorphic and f(1) = 1. Show that |1 − f(2)| ≤
1
3
|1 + f(2)|.
16. If f is entire and Re f ′(z) > 0 for all complex numbers z, prove that f is a polynomial of degree 1.
17. Given the polynomial P (z) = zn + an−1zn−1 + · · ·+ a0. Prove that max|z|=1 |P (z)| ≥ 1.
18. If f is an entire function mapping the unit circle into the unit circle (i.e. |f(z)| = 1 for |z| = 1), then
f(z) = eiθzn for some real θ and some positive integer n. [Hint: In the unit disk, f has finitely many
roots α1, . . . , αn, repeated according to multiplicities. Recall
∣∣∣∣ z − αj1− αjz
∣∣∣∣ = 1 for |z| = 1. Use the modulus
theorems to show f(z) = eiθ
n∏
j=1
z − αj
1− αjz first.]
19. Let f and g be holomorphic on a domain U . If fg is holomorphic on U , show that either f ≡ 0 or g is
a constant function.
20. Let f be holomorphic on the open unit disk D. Show that there is a sequence {zn} in D with |zn| → 1
such that {f(zn)} is a bounded sequence. [Hint: Consider the roots of f .]
44
21. Let w and z be in the open unit disk D. If f : D → D is holomorphic and f(w) = z, prove that
|f ′(w)| ≤ 1− |z|
2
1− |w|2 .
22. (Study’s Theorem) Let G be a convex domain and f be a bijective holomorphic function from the open
unit disk onto G. Prove that f(B(0, r)) is a convex domain for 0 < r < 1. (Recall a set S is convex if
w, z ∈ S implies tw + (1− t)z ∈ S for all t ∈ [0, 1].)
23. Let f : {z : |z| > 1} → C be holomorphic. Prove that if f is real-valued for all x ∈ (1,+∞), then f is
real-valued for all x ∈ (−∞,−1). [Hint: Use exercise 5 of chapter 6.]
45
13. Harmonic Functions and Conjugates
Definitions. A real-valued function u(x, y) having continuous second order partial derivatives on a region
D is harmonic on D iff it satisfies the Laplace equation
∂2u
∂x2
+
∂2u
∂y2
≡ 0 on D. If v is harmonic on D and
u+ iv is holomorphic on D, then we say v is a harmonic conjugate of u on D.
Theorem. If f = u+ iv is holomorphic on D, then u and v are harmonic on D.
Proof.By the Cauchy-Riemann equations, we get
∂2u
∂x2
=
∂
∂x
(
∂u
∂x
)
=
∂
∂x
(
∂v
∂y
)
=
∂
∂y
(
∂v
∂x
)
= − ∂
∂y
(
∂u
∂y
)
=
−∂
2u
∂y2
. Similarly,
∂2v
∂x2
= −∂
2v
∂y2
. QED
Remark. In general, the converse of the theorem is false. For example, let u(x, y) = ln
√
x2 + y2 = ln |z| on
C \{0}. If u = Re f for some holomorphic f there, then in B(c, ε) not containing 0, by the Cauchy-Riemann
equations, (f(z)−log z)′= ∂
∂x
Re(f(z)−log z)+i ∂
∂x
Im(f(z)−log z)= ∂
∂x
(u(z)−ln |z|)−i ∂
∂y
(u(z)−ln |z|)=0.
So the only possibility is f(z) = log z + constant, which is not continuous on C \{0}. However, for simply
connected domains, the converse of the theorem is true provided the Cauchy-Riemann equations are satisfied.
Theorem. If u is harmonic on a simply connected region D, then u is the real part of a holomorphic function
f on D.
Proof. Define g(z) =
∂u
∂x
(z) − i∂u
∂y
(z). Since u has continuous second order partial derivatives and the
Cauchy-Riemann equations
∂
∂x
(
∂u
∂x
)
=
∂
∂y
(
−∂u
∂y
)
,
∂
∂x
(
−∂u
∂y
)
= − ∂
∂y
(
∂u
∂x
)
are satisfied on D, by the
second theorem of chapter 6, g is holomorphic on D. By the polygon and integral theorem, g has an
antiderivative f on D, i.e. f ′(z) = g(z). Now
∂
∂x
(Re f − u) = Re fx − ∂u
∂x
= Re f ′ − ∂u
∂x
= 0 and
∂
∂y
(Re f − u) = Re fy − ∂u
∂y
= Re(if ′)− ∂u
∂y
= 0.
So Re f − u ≡ constant, i.e. u ≡ Re f + constant on D. QED
Below we shall give some rules for recovering a holomorphic function f = u+ iv from u (and v). Since
no proofs of these rules will be offered, proper checking should be made to ensure the functions obtained
have the correct real and imaginary parts.
Rule A. If u is harmonic on D containing an interval of the real axis, then f ′(z) =
∂u
∂x
(z, 0)− i∂u
∂y
(z, 0) and
f is obtained by integrating f ′.
Rule B. If u is harmonic on B(z0, r), z0 = x0 + iy0, then take f(z) = 2u
(
z + z0
2
,
z − z0
2i
)
− u(x0, y0).
Rule C. If u, v are harmonic conjugates on B(0, r), then take f(z) = u(z, 0) + iv(z, 0).
Examples. (1) u(x, y) = x2 − y2, ∂u
∂x
(x, y) = 2x =
∂v
∂y
(x, y),
∂u
∂y
(x, y) = −2y = −∂v
∂x
(x, y), which imply
v(x, y) = 2xy + constant.
46
Rule A: f ′(z) = 2z − i0⇒ f(z) = z2.
Rule B: Take z0 = 0, f(z) = 2u
(z
2
,
z
2i
)
= 2
(
z2
4
+
z2
4
)
= z2.
Rule C: Take v(x, y) = 2xy. Then f(z) = (z2 − 0) + i0 = z2.
(2) u(x, y) = e−y sinx,
∂u
∂x
(x, y) = e−y cos x =
∂v
∂y
(x, y),
∂u
∂y
(x, y) = −e−y sinx = −∂v
∂x
(x, y), which imply
v(x, y) = −e−y cosx+ constant.
Rule A: f ′(z) = cos z + i sin z = eiz ⇒ f(z) = −ieiz .
Rule B: Take z0 = 0, f(z) = 2e−z/2i sin
z
2
= e−z/2i
eiz/2 − e−iz/2
i
= −ieiz + i.
Rule C: Take v(x, y) = −e−y cosx. Then f(z) = sin z + i(− cos z) = −ieiz .
(3) u(x, y) =
x
x2 + y2
,
∂u
∂x
(x, y) =
y2 − x2
(x2 + y2)2
=
∂v
∂y
(x, y),
∂u
∂y
(x, y) =
−2xy
(x2 + y2)2
= −∂v
∂x
(x, y), which imply
v(x, y) = − y
x2 + y2
+ constant.
Rule A: f ′(z) = −z
2
z4
= − 1
z2
⇒ f(z) = 1
z
.
Rule B: Take z0 = 1, f(z) = 2
(z + 1)/2
(z + 1)2/4 + (z − 1)2/− 4 − 1 =
1
z
.
Rule C: Although u(0, 0) is undefined, we will try and see. Take v(x, y) = − y
x2 + y2
. Then f(z) =
1
z
+ 0 =
1
z
.
Mean-Value Theorem for Harmonic Functions. If u is harmonic on B(z0, R), then for 0 ≤ r < R,
u(z0) = Re f(z0) = Re
(
1
2pi
∫ 2pi
0
f(z0 + reiθ)dθ
)
=
1
2pi
∫ 2pi
0
Re f(z0 + reiθ)dθ =
1
2pi
∫ 2pi
0
u(z0 + reiθ)dθ.
Maximum/Minimum Principle for Harmonic Functions. If u is a nonconstant harmonic function on
a region D, then u has no maximum or minimum on D. (If D is bounded and u is also continuous on D and
its boundary ∂D, then u attains maximum and minimum values on ∂D only.)
Proof. Suppose u has a maximum or minimum at p ∈ D. Let D0 = B(p, ε) be a disk in D. Then u has a
maximum or minimum at p onD0. Now u = Re f onD0 for some nonconstant holomorphic function f onD0.
By the open mapping theorem, f(D0) is open in C. Hence Re f(D0) = {Re f(z): z ∈ D0} = {u(z): z ∈ D0}
is open in R. Then u cannot have a maximum or minimum on D0 ⊆ D, a contradiction. QED
Corollary. If u1, u2 are harmonic on a bounded region D, continuous on D and its boundary ∂D, u1 = u2
on ∂D, then (considering u = u1 − u2, we get) u1 ≡ u2 on D. So, the function u 7→ u|∂D is bijective.
Poisson Integral Formula. If u is harmonic on B(0, 1) and continuous on B(0, 1), then for z = reiθ ∈
B(0, 1), u(z) =
1
2pi
∫ 2pi
0
1− r2
1− 2r cos (θ − t) + r2u(e
it) dt. This formula gives an inverse to the function u 7→
u|∂B(0,1).
Proof. Since B(0, 1) is simply connected, u = Re f for some holomorphic function f on B(0, 1). For
|z| = r < R < 1, observe that z ∈ B(0, R), but R2/z 6∈ B(0, R). By the Cauchy integral formula,
f(z) =
1
2pii
∫
|w|=R
f(w)
w − z dw −
1
2pii
∫
|w|=R
f(w)
w − R
2
z
dw
︸ ︷︷ ︸
= 0
=
1
2pii
∫
|w|=R
(
1
w − z −
1
w − R
2
z
)f(w) dw.
47
For w = Reit and z = reiθ, we have dw = iReit dt and
1
w − z −
1
w − R
2
z
=
1
Reit − reiθ −
1
Reit − R
2
r
eiθ
=
(r − R
2
r
)eiθ
(Reit − reiθ)(Reit − R
2
r
eiθ)
=
R2 − r2
(Rei(t−θ) − r)(Rei(θ−t) − r)Reit =
R2 − r2
R2 − 2Rr cos(θ − t) + r2
1
Reit
.
So, f(z) =
1
2pi
∫ 2pi
0
R2 − r2
R2 − 2Rr cos (θ − t) + r2 f(Re
it) dt. Taking the real part of both sides and letting
R→ 1−, we get u(z) = 1
2pi
∫ 2pi
0
1− r2
1− 2r cos (θ − t) + r2u(e
it) dt. QED
Remark. In an advanced course, it can be shown that if u is continuous on C(0, 1), then u(z), defined by
the Poisson integral above, is harmonic on B(0, 1) and continuous on B(0, 1).
Exercises
1. If f = u+ iv is holomorphic, prove that u+ v and uv are harmonic.
2. If u is harmonic, prove that all the partial derivatives of u are harmonic, but u2 is not harmonic, unless
u is constant.
3. Find holomorphic functions whose real parts are
(a) x3y − xy3; (b) ex2−y2 cos(2xy);
(c) arctan
( y
x
)
; (d)
1− x2 − y2
(1− x)2 + y2 .
4. Is the parenthetical part of the maximum/minimum principle true if D is unbounded? [Hint: Consider
u(x, y) = ±y on {z : Im z ≥ 0}.]
5. Using the Poisson integral formula, find
∫ 2pi
0
dt
5− 4 cos t and
∫ 2pi
0
3ecos t cos(sin t)
5− 4 cos(t−√2) dt.
6. Show that in polar coordinates, Laplace’s equation is
1
r
∂
∂r
(
r
∂u
∂r
)
+
1
r2
∂2u
∂θ2
= 0.
7. (a) Suppose g is holomorphic on the closed unit disk, g(0) = 1 and Re g(z) > 0 for |z| < 1. Use the
Poisson integral formula to show that Re g(z) ≤ 1 + |z|
1− |z| for |z| < 1.
(b) Suppose f : {z : |z| ≤ 1} → {z : 0 < |z| < 1} is holomorphic. Show that |f(z)| ≤ |f(0)|
1+|z|
1−|z| for
|z| < 1.
48
14. Morera's Theorem
The rectangle theorem asserts that if f is holomorphic, then f has the rectangle property. Below we shall
prove the converse for continuous functions. Thus, holomorphicity is equivalent to the rectangle property.
Morera’s Theorem. If a continuous function f has the rectangle property on an open set U , then f is
holomorphic on U .
Proof. For a ∈ U , take a disk B(a, r) in U . Since f has the rectangle property, the integral theorem implies
f has an antiderivative F on B(a, r). That is, F is differentiable on B(a, r) and F ′ = f . Then F is infinitely
differentiable, so f is differentiable on B(a, r). Since a is arbitrary, f is holomorphic on U . QED
Corollary. If f is continuous on a region D and holomorphic on D\L, where L is a (horizontal) line segment
(or in the extreme case, a point), then f is holomorphic on D (by the Extension Theorem).
Example. Define F (z) =
∫ ∞
0
ezt
t + 1
dt for U = {z: Re z < 0}. (Observe that
∫ ∞
0
∣∣∣∣ eztt+ 1
∣∣∣∣ dt ≤ ∫ ∞
0
ext dt
= −1
x
=
1
|x| for x = Re z < 0. So, F is defined on U .) Let Γ be a “rectangle” in U . For fixed t, gt(z) =
ezt
t+ 1
is holomorphic on U. By Cauchy’s theorem,
∫
Γ
gt(z) dz = 0. Then
∫
Γ
F (z) dz =
∫
Γ
∫ ∞
0
ezt
t+ 1
dt dz =
∫ ∞
0
∫
Γ
ezt
t+ 1
dz dt =
∫ ∞
0
∫
Γ
gt(z) dz dt = 0
(The interchange of integrals is valid because
∫
Γ
∫ ∞
0
∣∣∣∣ eztt + 1
∣∣∣∣ dt |dz| ≤ ∫
Γ
1
|x| |dz| <∞, since |x| = |Re z| is
bounded below by the positive distance from Γ to 0.) So F is holomorphic on U by Morera’s theorem.
As an application of Morera’s theorem, we will consider limits of holomorphic functions. In general, the
limit of a sequence of continuous functions may not be continuous (e.g. fn(x) = xn, 0 ≤ x ≤ 1 has the limit
f(x) =
{ 0 if 0 ≤ x < 1
1 if x = 1
, which is not continuous). So the limit of a sequence of holomorphic functions may
not be holomorphic. However, if the convergence to the limit is uniform, then it is holomorphic. In that
case, the derivatives will also converge uniformly to the derivative of the limit.
Weierstrass’ Theorem. If fn is a sequence of holomorphic functions on a region D and fn converges
uniformly to f on every closed disk B(a, r) = {z: |z − a| ≤ r} in D, then f is holomorphic on D. (Fur-
thermore, for any positive integer k, f (k)n will also converge uniformly to f (k) on every closed disk in D, i.e.
lim
n→∞
dk
dzk
fn =
dk
dzk
lim
n→∞ fn uniformly on closed disks in D.)
Proof. For every a in D, take a closed disk B(a, r) in D. Since the fn’s are continuous and the convergence
is uniform, f is continuous on B(a, r). For a “rectangle” Γ in B(a, r),
∫
Γ
f(z) dz =
∫
Γ
lim
n→∞ fn(z) dz =
lim
n→∞
∫
Γ
fn(z) dz = 0. By Morera’s theorem, f is holomorphic at a for any a ∈ D.
(For the parenthetical statement, let k be a positive integer, B(a, r) ⊆ D and ε > 0 be given. Then
there is R > r such that B(a, r) ⊆ B(a,R) ⊆ D. Fix a ρ such that 0 < ρ < R − r. Since fn converges
49
uniformly to f on B(a,R), there is N such that n ≥ N ⇒ |fn(w) − f(w)| < ε for all w ∈ B(a,R). Now for
any arbitrary z ∈ B(a, r), C(z, ρ) ⊆ B(a,R). By the Cauchy integral formula for derivatives and the M -L
inequality,
∣∣∣f (k)n (z) − f (k)(z)∣∣∣ =
∣∣∣∣∣ k!2pii
∫
|w−z|=ρ
fn(w) − f(w)
(w − z)k+1 dw
∣∣∣∣∣ ≤ k!2pi
(
1
ρk+1
ε
)
2piρ =
k!ε
ρk
.
Since z is arbitrary in B(a, r), uniform convergence is proved.) QED
The next application of Morera’s theorem concerns extending analyticity across a line segment.
Schwarz Reflection Principle. Let D be a symmetric region with respect to the real axis, Dupper = {z ∈
D : Im z > 0} and Dlower = {z ∈ D : Im z < 0}. If f is continuous on Dupper ∪ (D ∩ R), holomorphic
on Dupper and real-valued on D ∩ R, then f can be extended to a holomorphic function on D by defining
f(z) = f(z) for z ∈ Dlower.
D upper
D lower
R
Proof. The hypothesis that f is real-valued on D ∩R is equivalent to f(x) = f(x)
for all x ∈ D ∩R. This makes the extension continuous on all of D. On Dlower, we
can apply the definition of derivative to get
f ′(z) = lim
h→0
f(z + h)− f(z)
h
= lim
h→0
f(z + h)− f(z)
h
= f ′(z)
for all z ∈ Dlower. So, f is holomorphic on D \ R . By the corollary to Morera’s
theorem, f is holomorphic on D.
QED
Exercises
1. Use Morera’s Theorem to show that f(z) =
∫ 1
0
sin zt
t
dt is an entire function.
2. Prove that if f is continuous on the closed unit disk {z : |z| ≤ 1}, holomorphic on the open unit disk
{z : |z| < 1} and real-valued on the unit circle {z : |z| = 1}, then f is a constant function. [Hint:
Consider f ◦ T, where T (z) = (i − z)/(i + z), which sends the upper half plane to the open unit disk.]
3. Let f be an entire function which is real on the real axis.
(a) Use Schwarz reflection principle to prove that f(z) = f(z) for every complex number z.
(b) If f is also imaginary on the imaginary axis, prove that f is an odd function by considering the
power series of f at the origin.
4. Suppose f is holomorphic on D = {z : |z| ≤ 1, Im z > 0} and is continuous on ∂D. Prove that it is
impossible for f(x) = |x| for all x ∈ R .
5. Define f : C \[0, 1]→ C by f(z) =
∫ 1
0
√
t
t− z dt. Prove that f is holomorphic.
50
15. Isolated Singularities
In this section, we will consider a two-sided infinite series representation near a singular point in the
domain of a holomorphic function.
Definition.We say
∞∑
k=−∞
wk converges to L iff
−1∑
k=−∞
wk,
∞∑
k=0
wk converge and
−1∑
k=−∞
wk +
∞∑
k=0
wk = L.
Laurent Series Representation. If f is holomorphic on the annulus A = {z: 0 ≤ R1 < |z−a| < R2 ≤ ∞},
then for all z ∈ A, f(z) =
∞∑
k=−∞
ak(z − a)k, where ak = 12pii
∫
|w−a|=r
f(w)
(w − a)k+1 dw for any r ∈ (R1, R2).
(Also the convergence is absolute on A, uniform on any smaller annulus {z:R1 < R′1 ≤ |z − a| ≤ R′2 < R2}
and the coefficients ak are unique for A .) The series is called the Laurent series of f(z) on A.
R1
r1 r2 R2
z0
simply
connected
upper half
Proof. By a change of variable, we may assume a = 0. Let R1 < r1 < r2 < R2.
Also, let C1, C2 be the counterclockwise circles |z| = r1, |z| = r2. For r1 < |z0| <
r2, the function g(z) =
 f(z) − f(z0)z − z0 if z 6= z0, z ∈ A
f ′(z0) z = z0
is holomorphic on A.
Taking the line segments [r1, r2] and [−r2,−r1] as cross-cuts, we get by
Cauchy’s theorem,
∫
C2−C1
g(w) dw = 0. Using the definition of g(z) and grouping
terms, we get∫
C2
f(w)
w − z0 dw−
∫
C1
f(w)
w −w0dw = f(z0)
(∫
C2
dw
w − z0 −
∫
C1
dw
w − z0
)
= 2piif(z0).
For w ∈ C2, |z0| < |w| = r2, 1
w − z0 =
1
w(1− z0
w
)
=
1
w
(
1 +
z0
w
+
(z0
w
)2
+ . . .
)
=
∞∑
k=0
zk0
wk+1
.
For w ∈ C1, |w| = r1 < |z0|, 1
w − z0 =
−1
z0(1− w
z0
)
= − 1
z0
(
1 +
w
z0
+
(
w
z0
)2
+ . . .
)
= −
−1∑
k=−∞
zk0
wk+1
.
Both series converge uniformly in w because
∣∣∣z0
w
∣∣∣ < 1 on C2 and ∣∣∣∣wz0
∣∣∣∣ < 1 on C1. So, for R1 < r < R2,
2piif(z0) =
∞∑
k=0
∫
C2
f(w)zk0
wk+1
dw︸ ︷︷ ︸
I
+
−1∑
k=−∞
∫
C1
f(w)zk0
wk+1
dw︸ ︷︷ ︸
II
=
∞∑
k=−∞
( ∫
|w|=r
f(w)
wk+1
dw
)
zk0 .
(Observe that
∫
C2
=
∫
|w|=r
=
∫
C1
by using cross-cuts as above.) Letting r1 → R+1 and r2 → R−2 , we get
the result. (For the parenthetical statement, we observe that the series I and II are power series in z0 and
z−10 , respectively. Hence, the absolute and uniform convergence properties and the uniqueness of coefficients
follow from the corresponding properties for power series.) QED
51
An important point of the theorem is that the Laurent series coefficients for a holomorphic function
on an annulus are unique. (To elaborate on this, suppose f(z) =
∞∑
k=−∞
bk(z − a)k uniformly on the same
annulus, then
1
2pii
∫
|w−a|=r
f(w)
(w − a)n+1 dw =
∞∑
k=−∞
bk
(
1
2pii
∫
|w−a|=r
(w − a)k−n−1 dw
)
= bn. Therefore, the
coefficients bk’s are the same as the coefficients ak’s in the Laurent series representation above.) So in
computing Laurent series, we do not have to compute the coefficients by integrals. Instead, we can apply
other methods (such as the formula for summing geometric series or power series of common functions,)
depending on the given functions and annuli.
Example. Find the Laurent series of f(z) =
1
z2(1 − z) for A1 = {z: 0 < |z| < 1}, A2 = {z: 1 < |z| < ∞}
and A3 = {z: 0 < |z − 1| < 1}.
Solution. On A1,
1
z2(1− z) =
1
z2
(1 + z + z2 + . . .) =
1
z2
+
1
z
+ 1 + z + z2 + . . ..
On A2,
∣∣∣∣1z
∣∣∣∣ < 1, 1z2(1 − z) = − 1z3
(
1
1− 1z
)
= − 1
z3
(1 +
1
z
+
1
z2
+
1
z3
+ . . .) = . . .− 1
z6
− 1
z5
− 1
z4
− 1
z3
.
On A3, the series is of the form
∞∑
k=−∞
ak(z−1)k. We make use of the fact |z−1| < 1 and write z = 1+(z−1), so
1
z2(1− z) =
1
(1 + (z − 1))2 ·
1
(1− z) . For w = z−1, |w| < 1,
1
(1 + w)2
= − d
dw
(
1
1 +w
)
= −
∞∑
n=0
d
dw
(−w)n =
∞∑
n=1
(−1)n+1nwn−1, so
1
z2(1− z) = −
1
(z − 1) [1− 2(z − 1) + 3(z − 1)
2 − 4(z − 1)3 + . . .] = − 1
z − 1 + 2− 3(z − 1) + 4(z − 1)
2 − . . . .
Definitions. If f(z) is holomorphic in a deleted neighborhood of z0 (i.e. on some B(z0, ε) \ {z0}), then f(z)
is said to have an isolated singularity at z0. The Laurent expansion of f(z) at z0 is the Laurent series of f(z)
on A = {z: 0 < |z − z0| < ε}. If it is of the form
(i) a0 + a1(z − z0) + a2(z − z0)2 + . . ., we say z0 is a removable singularity of f(z) (in this case, we can
define f(z0) = a0 and f(z) becomes holomorphic at z0 because the Laurent series is the same as the
power series of f(z) in B(z0, ε));
(Example: f(z) =
sin z
z
has a removable singularity at 0. For 0 < |z| < ε, sin z
z
=
z − z
3
3!
+
z5
5!
− . . .
z
= 1 + 0z − z
2
3!
+ 0z3 +
z4
5!
− . . .. So defining f(0) = 1 will make f(z) holomorphic at 0.)
(ii)
a−k
(z − z0)k +
a−k+1
(z − z0)k−1 + . . .+
a−1
(z − z0) + a0 + a1(z − z0) + . . . with a−k 6= 0, we say z0 is a pole of
order k for f(z) (in this case, F (z) = (z − z0)kf(z) is holomorphic at z0, F (z0) = a−k 6= 0 and
lim
z→z0
f(z) = lim
z→z0
F (z)
(z − z0)k =∞);
(Example: f(z) =
1
sin z
has a pole (of order 1) at 0. For 0 < |z| < ε,
1
sin z
=
1
z(1− z
2
3!
+
z4
5!
− . . .)
=
1
z
(1 +
z2
6
+
7
360
z4 + . . .) =
1
z
+ 0 +
1
6
z + 0z2 +
7
360
z3 + . . . .)
52
(iii) . . .+
a−k
(z − z0)k +
a−k+1
(z − z0)k−1 + . . .+
a−1
z − z0 + a0 + a1(z − z0) + . . ., we say z0 is an essential singularity
of f(z).
(Example: f(z) = sin
1
z
has an essential singularity at 0. For 0 < |z| < ε,
sin
1
z
=
1
z
− 1
3!
1
z3
+
1
5!
1
z5
− . . . = . . .+ 1
120z5
+
0
z4
− 1
6z3
− 0
z2
+
1
z
+ 0 + 0z + . . . .)
Definitions. If the Laurent series of f(z) on 0 < |z− z0| < ε is
∞∑
k=−∞
ak(z − z0)k, then
−1∑
k=−∞
ak(z − z0)k is
the principal part of f at z0 and
∞∑
k=0
ak(z − z0)k is the holomorphic part of f at z0.
Theorem. Let z0 be an isolated singularity of f(z).
(i) (Riemann’s principle.) If lim
z→z0
(z − z0)f(z) = 0 (e.g. |f(z)| ≤ M near z0), then z0 is a removable
singularity.
(ii) If there is a positive integer k such that lim
z→z0
(z − z0)kf(z) 6= 0, but lim
z→z0
(z − z0)k+1f(z) = 0, then z0 is
a pole of order k.
Proof. (i) The function g(z) =
{
(z − z0)f(z) if z 6= z0,
0 if z = z0,
is holomorphic near z0 and continuous at z0.
By the extension theorem, g is holomorphic at z0. So, near z0, f(z) =
g(z)
z − z0 =
∞∑
k=0
g(k+1)(z0)
(k + 1)!
(z − z0)k.
Therefore, z0 is a removable singularity of f .
(ii) By (i), lim
z→z0
(z − z0)[(z − z0)kf(z)] = 0 implies (z − z0)kf(z) has a removable singularity at z0. So,
near z0, (z − z0)kf(z) =
∞∑
n=0
cn(z − z0)n. Now c0 = lim
z→z0
(z − z0)kf(z) 6= 0. Therefore, near z0, f(z) =
c0
(z − z0)k +
c1
(z − z0)k−1 + · · ·, i.e. f has a pole of order k at z0. QED
Casorati-Weierstrass Theorem. If z0 is an essential singularity of f(z), then the range set f(D∗) =
{f(z): 0 < |z − z0| < ε} is dense in C for every ε > 0. (Here “dense” means in every open disk in C,
regardless of size and location, we can find a point of f(D∗).)
Proof. Suppose f(D∗) is not dense in C for some ε > 0. Then f(D∗) is disjoint from some disk B(a, r),
i.e. |f(z) − a| ≥ r for 0 < |z − z0| < ε. It follows that g(z) = 1
f(z) − a is bounded (by
1
r
) near z0. By
Riemann’s principle, z0 is a removable singularity of g(z), i.e. g can be defined at z0 so as to be holomorphic
there. Now f(z) = a+
1
g(z)
. If g(z0) 6= 0, then lim
z→z0
(z − z0)f(z) = 0, which forces f to have a removable
singularity at z0, a contradiction. If g(z0) = 0, then g(z) = (z − z0)kh(z) for some positive integer k and
some holomorphic function h(z) near z0 with h(z0) 6= 0. It follows that lim
z→z0
(z − z0)kf(z) = 1
h(z0)
6= 0 and
lim
z→z0
(z − z0)k+1f(z) = 0. Then f has a pole of order k at z0, a contradiction. QED
In an advanced course, a theorem called the Great Picard Theorem is usually proved, which asserts that
f(D∗) can miss at most one complex number and in fact, the equation f(z) = c has infinitely many solutions
in D∗, except perhaps for one complex value c. (For example, 0 is an essential singularity of e1/z and in every
neighborhood of 0, the range of e1/z is C\{0}.) This is stronger than the Casorati-Weierstrass theorem.
53
Summary. Let z0 be an isolated singularity of f . Then
(i) z0 is a removable singularity of f ⇐⇒ lim
z→z0
f(z) ∈ C ⇐⇒ f(z) is bounded near z0 (i.e. |f(z)| ≤ M
near z0 for some M ) ⇐⇒ lim
z→z0
(z − z0)f(z) = 0;
(ii) z0 is a pole of f ⇐⇒ lim
z→z0
f(z) = ∞; (also, z0 is a pole of order k ⇐⇒ lim
z→z0
(z − z0)kf(z) 6= 0 and
lim
z→z0
(z − z0)k+1f(z) = 0;)
(iii) z0 is an essential singularity of f ⇐⇒ f(D∗) = {f(z): 0 < |z − z0| < ε} is dense in C for any
ε > 0 ⇐⇒ lim
z→z0
f(z) doesn’t exist.
(Reasons.
(i) If z0 is a removable singularity, then near z0, f(z) =
∞∑
k=0
ak(z − z0)k, so lim
z→z0
f(z) = a0 ∈ C.
If lim
z→z0
f(z) = A ∈ C, then for each ε > 0, there is δ > 0 such that 0 < |z − z0| < ε ⇒ |f(z) − A| < δ.
So |f(z)| < A + δ in 0 < |z − z0| < ε.
If |f(z)| ≤M near z0, then |(z − z0)f(z)| ≤ M |z − z0| → 0 as z → z0.
If lim
z→z0
(z − z0)f(z) = 0, then z0 is a removable singularity by Riemann’s Principle.
(ii) If z0 is a pole (of order k), then from the definition, we have lim
z→z0
f(z) =∞ (and lim
z→z0
(z− z0)kf(z) 6= 0,
lim
z→z0
(z − z0)k+1f(z) = 0).
If lim
z→z0
f(z) =∞, then z0 is not a removable singularity by (i) above and it is not an essential singularity
by Casorati-Weierstrass Theorem (because the range of f omits small values near z0). (The converse
part of the parenthetical statement was proved in a theorem earlier.)
(iii) If z0 is an essential singularity, then f(D∗) is dense in C for any ε > 0 by Casorati-Weierstrass Theorem.
If f(D∗) is dense in C, then lim
z→z0
f(z) cannot exist.
If lim
z→z0
f(z) doesn’t exist, then by (i) and (ii), z0 is not a removable singularity or a pole, hence it must
be an essential singularity.)
Exercises
1. Identify the isolated singularities of the following functions and classify each as a removable singularity,
a pole (and its order) or an essential singularity:
(a)
1
z4 + z2
; (b) cot z;
(c)
e1/z
2
z − 1 ; (d)
z2 − 1
sinpiz
.
2. (i) Find the Laurent series of
z + 1
z − 1 on (a) |z| < 1 and (b) |z| > 1.
(ii) Find the Laurent series of
1
z2 − 4 on (a) 0 < |z − 2| < 4 and (b) 2 < |z| <∞.
(iii) Find the Laurent series of
1
(z2 − 1)2 on 1 < |z| < 2.
(iv) Find the first four terms of the Laurent series of
ez
z(z2 + 1)
on 0 < |z| < 1.
3. Find
∫
|z|=r
sin
1
z
dz for positive r 6= 1
npi
. As usual, the circle |z| = r is given the counterclockwise
orientation.
54
4. Suppose f is holomorphic on C \{0} and satisfies |f(z)| ≤
√|z| + 1√|z| . Prove that f is a constant
function.
5. Find the roots of sinh z. What is the largest r for which there exist c0, c1, c2, . . . ∈ C such that
(z2 + pi2)(ez − 1)
sinh z
=
∞∑
n=0
cnz
n for |z| < r?
6. Let G be a region and f : G→ C be continuous. If f2 is holomorphic on G, show that f is holomorphic
on G. [Hint: First show f is holomorphic at z such that f(z) 6= 0. Then consider the singularity type
of the roots of f .]
7. If f has a pole at 0, show that ef cannot have a pole at 0.
8. Let f be holomorphic on {z : R < |z| <∞}. We say ∞ is a removable singularity, a pole of order k or
an essential singularity of f(z) if and only if 0 is a removable singularity, a pole of order k or essential
singularity of f
(
1
z
)
, respectively.
(a) Prove that an entire function with a pole at ∞ is a polynomial.
(b) Prove that a holomorphic function on C∪{∞} except for isolated poles must be a rational function.
9. Prove that the image of the plane under a nonconstant entire mapping f is dense in the plane. [Hint:
If f is not a polynomial, then consider f
(
1
z
)
.]
10. Can the positive integers {1, 2, 3, . . .} be partitioned into a finite number of sets S1, S2, . . . , Sk, each of
which is an arithmetic progression, i.e.
S1 = {a1, a1 + d1, a1 + 2d1, . . .}
S2 = {a2, a2 + d2, a2 + 2d2, . . .}
· · ·
Sk = {ak, ak + dk, ak + 2dk, . . .},
and such that there are no equal common differences (i.e. di 6= dj for i 6= j)? [Hint: Consider the series∑
n∈Sj
zn for j = 1, 2, . . ., k.] (This exercise is due to D. J. Newman.)
11. Prove that there is no bijective conformal mapping from G = {z : 0 < |z| < 1} to A = {z : 1 < |z| < 2}.
[Hint: Removable singularity and open mapping theorem.]
55
16. Residues and Roots
Definition. Let
∞∑
k=−∞
ak(z − z0)k be the Laurent series of f(z) on 0 < |z− z0| < ε, then the residue of f(z)
at z0 is Res
z=z0
f(z) = Res (f, z0) = a−1.
Observe that if f(z) =
∞∑
k=−∞
ak(z − z0)k for 0 < |z − z0| < ε, then for 0 < r < ε,
∫
|z−z0|=r
f(z) dz
=
∞∑
k=−∞
ak
∫
|z−z0|=r
(z − z0)k dz = 2piia−1 because (z − z0)k has antiderivative (z − z0)
k+1
k + 1
for k 6= −1. This
is the reason the word “residue” is used (because a−1 is the only coefficient that remains).
Definition. A function f is meromorphic on a region D iff f is holomorphic on D except for (isolated) poles.
Residue Theorem. Suppose f is meromorphic on a simply connected region D with poles at z1, . . . , zn.
Let Γ be a closed curve on D not passing through z1, . . . , zn, then∫
Γ
f(z) dz = 2pii
n∑
j=1
n(Γ, zj) Res
z=zj
f(z).
Proof. Let Pj
(
1
z − zj
)
=
−1∑
k=−mj
ak,j(z − zj)k be the principal part of f at zj , where mj is the order of the
pole at zj . Then the function g(z) = f(z) −
n∑
j=1
Pj
(
1
z − zj
)
is holomorphic on D. (At zj , the singularity
is cancelled.) By Cauchy’s theorem,
0 =
∫
Γ
g(z) dz =
∫
Γ
f(z) dz −
n∑
j=1
∫
Γ
−1∑
k=−mj
ak,j(z − zj)k dz =
∫
Γ
f(z) dz − 2pii
n∑
j=1
a−1,j n(Γ, zj).
QED
Remark. The residue theorem is true even if there are essential singularities. The proof for that amounts
to replacing Γ by a sequence of small circles about the singularities.
Cauchy Integral Formula for Derivatives. If f is holomorphic on a simply connected region D and a is
a point not on a closed curve Γ in D, then
∫
Γ
f(w)
(w − a)n+1 dw = 2pii n(Γ, a)
f (n)(a)
n!
.
Proof. Near a, f(z) = f(a) + f ′(a)(z − a) + . . .+ f
(n)(a)
n!
(z − a)n + . . .. By the residue theorem,∫
Γ
f(w)
(w − a)n+1 dw = 2pii n(Γ, a)Resz=a
f(z)
(z − a)n+1 = 2pii n(Γ, a)
f (n)(a)
n!
.
QED
56
Argument Principle. If f is meromorphic on a simply connected region D and Γ is a simple closed curve
in D not passing through the roots zj nor the poles pk of f , then
1
2pii
∫
Γ
f ′(z)
f(z)
dz = n−m, where n and m
are the number of roots and poles (counting multiplicities) of f inside Γ, respectively.
p
1
p
2
z 1
z 2
z 3
z 4
Γ
0
f
f Γ
(If we let w = f(z), then n−m = 1
2pii
∫
Γ
f ′(z)
f(z)
dz =
1
2pii
∫
f◦Γ
dw
w
= n(f ◦ Γ, 0)
=
∆Γ arg f(z)
2pi
, where ∆Γ arg f(z) is the change of argument in f(z) along Γ.
This is the reason for the name Argument Principle.)
Proof. The isolated singularities of
f ′
f
are at the zj’s and pk’s. If the order of zj as a root of f is nj, then
f(z) = anj (z − zj)nj + . . . near zj and
f ′(z)
f(z)
=
nj
z − zj + . . . near zj. If the order of pk as a pole of f is mk,
then f(z) =
a−mk
(z − pk)mk + . . . near pk and
f ′(z)
f(z)
=
−mk
z − pk + . . .. By the residue theorem,
1
2pii
∫
Γ
f ′(z)
f(z)
dz =
∑
j
Res
z=zj
f ′(z)
f(z)
+
∑
k
Res
z=pk
f ′(z)
f(z)
=
∑
j
nj +
∑
k
(−mk) = n−m.
QED
Examples. (1) Find the number of roots of f(z) = 17z5 + z2 − 1 in each quadrant and on the real or
imaginary axes.
iR
-iR
γ 1
γ 2
- 1 = f
f ( iR
f (
-iR
)
(0)
)
Solution.On R, f ′(x) = 0 for x = 0 and − 3
√
2/85. From f(0) = −1 < f(− 3√2/85) < 0,
a sketch of the graph of y = f(x) shows there is one positive root only. Since f(iy) =
−y2 − 1 + 17y5i 6= 0 for y real, there is no imaginary root.
We now apply the argument principle to the right half plane. Consider Γ to be
the boundary of the half disk of radius R (R huge) in the right half plane. On γ1,
z = Reiθ, −pi
2
≤ θ ≤ pi
2
, because R is huge, f(z) ≈ 17z5 = 17R5ei5θ. This implies
∆γ1 arg f(z) ≈ 5(
pi
2
− (−pi
2
)) = 5pi.
On γ2, z = iy, where y decreases from R to −R, f(iy) = −y2 − 1︸ ︷︷ ︸
<0
+17y5i. So
arg f(iR) = arctan
17R5
−R2 − 1 →
pi
2
+
, arg f(−iR) = arctan −17R
5
−R2 − 1 →
3pi
2
−
asR→ +∞.
These imply ∆γ2 arg f(z) = arg f(−iR) − arg f(iR) ≈
3pi
2
− pi
2
= pi.
So ∆Γ arg f(z) = 5pi + pi = 6pi and there are
6pi
2pi
= 3 roots inside Γ when R is huge (one of which is real
as shown above). Therefore there is one positive real root and two pairs of complex roots in conjugate (one
complex root in each quadrant).
(2) Let w ∈ C \[−2, 2] and k a positive integer. Define f(z) = zk + 1
zk
. Show that f takes the value w
exactly k times in D = {z: |z| < 1} (i.e. f(z) = w has exactly k solutions in D).
Solution. Let g(z) = f(z) − w. We want to show g has exactly k roots in D. Let Γ be the unit circle
z(θ) = eiθ, 0 ≤ θ ≤ 2pi, g(eiθ) = eikθ + e−ikθ − w = 2 cos kθ −w.
57
ww
w
-2 - 2 -
Then g ◦ Γ is the line segment joining −2 − w to 2 − w. Since w 6∈ [−2, 2], g ◦ Γ does
not wind around 0, so by the argument principle,
0 = n(g ◦ Γ, 0) =
(
number of roots
of g(z) in D
)
−
(
number of poles
of g(z) in D
)
.
Since 0 is the only pole of f(z) (hence also of g(z)) in D and it is of order k, so the number of roots of g(z)
in D is k. Therefore, f(z) = w has exactly k solutions in D.
Rouche´’s Theorem (Estermann-Glicksberg Form). If f and g are holomorphic inside and on a simple
closed curve Γ and |f(z) + g(z)| < |f(z)|+ |g(z)| for all z ∈ Γ, then f and g have the same number of roots
inside Γ. (Note the inequality on Γ implies f and g have no roots on Γ.)
Proof. For z ∈ Γ, |f(z) + g(z)| < |f(z)| + |g(z)| implies f(z), g(z) 6= 0 and f(z)
g(z)
is not zero nor a positive
real number (otherwise |f(z)+g(z)| = |f(z)|+ |g(z)|). Then log f(z)
g(z)
can be defined with 0 < arg
f(z)
g(z)
< 2pi.
So 0 =
∫
Γ
(
log
f(z)
g(z)
)′
dz=
∫
Γ
f ′(z)
f(z)
dz−
∫
Γ
g′(z)
g(z)
dz and the result follows from the argument principle. QED
Remark. To show f and g have the same number of roots inside Γ, it is sufficient to show |f(z)+g(z)| < |f(z)|
or |f(z) + g(z)| < |g(z)|.
Examples. (1) Show that all the roots of f(z) = z5 + 3z + 1 are in the disk |z| < 2.
Solution. Let g(z) = −z5, which has 5 roots (counting multiplicities) in the disk |z| < 2. Let Γ be the circle
|z| = 2, then for z ∈ Γ, |f(z) + g(z)| = |3z + 1| ≤ 3|z|+ 1 = 7 < |g(z)| = |z|5 = 25 = 32. So by Rouche´’s
theorem, f has 5 roots inside |z| = 2.
(2) What is the smallest positive integer r such that f(z) = z5 + 48z + 64 has a root in the disk |z| < r?
Solution. For |z| = 1, let g(z) = −64, then |f(z) + g(z)| = |z5 + 48z| ≤ |z|5 + 48|z| = 49 < 64 = |g(z)|, so
by Rouche´’s theorem, f has no roots inside |z| = 1.
For |z| = 2, let g(z) = −48z, then |f(z) + g(z)| = |z5 + 64| ≤ |z|5 + 64 = 96 = 48|z| = |g(z)| ≤
|f(z)|+ |g(z)|. For equality to hold throughout, we must have z5 = 32 and f(z) = 0. Then 32+48z+64 = 0
implies z = −2, which contradicts z5 = 32. So one of the inequalities is strict, i.e. on |z| = 2, |f(z)+ g(z)| <
|f(z)|+ |g(z)|. By Rouche´’s theorem, f has one root inside |z| = 2. Therefore r = 2.
(3) If a1, . . . , an, b ∈ D = {z: |z| < 1} and f(z) =
n∏
j=1
z − aj
1− ajz , then the equation f(z) = b has exactly n
solutions (counting multiplicities) in D.
Solution. Let g(z) = b−f(z), then for |z| = 1, |f(z)+g(z)| = |b| < 1 =
n∏
j=1
∣∣∣∣ z − aj1− ajz
∣∣∣∣ = |f(z)|. By Rouche´’s
theorem, g has n roots in D, since f has n roots a1 . . . , an in D. So f(z) = b has n solutions in D.
Local Mapping Theorem. Let f be nonconstant holomorphic on open Ω ⊆ C and z0 be a root of f of
order m in Ω. Then for ε > 0 small (i.e. B(z0, ε) \ {z0} has no root of f or f ′) , there is a δ > 0 such that
the equation f(z) = w with 0 < |w| < δ has exactly m distinct solutions (each of order 1) in B(z0, ε), i.e.
near z0, f is locally a m-to-1 mapping.
Proof. Since f is nonconstant, by the identity theorem, there cannot be any sequence of roots of f (or f ′)
converging to z0. So we may pick ε > 0 small so that f(z) 6= 0 and f ′(z) 6= 0 for z ∈ B(z0, ε) \{z0} ⊆ Ω. Let
δ = min
|z−z0|=ε
|f(z)|, then δ > 0. Suppose 0 < |w| < δ, then for |z − z0| = ε, we have |f(z) + (w − f(z)) | =
58
|w| < δ ≤ |f(z)|+ |w− f(z)|. By Rouche´’s theorem, f and w− f have the same number of roots in B(z0, ε),
namely m. Since f ′(z) 6= 0 for z ∈ B(z0, ε), the solutions of f(z) = w are distinct and of order 1. QED
Inverse Mapping Theorem. If f is univalent (i.e. injective and holomorphic) on open U ⊆ C, then
f−1: f(U )→ U is holomorphic.
Proof. (Since f is injective, f is nonconstant. Hence by the open mapping theorem, f(U ) is open.) For an
arbitrary a ∈ U , we will show that f−1 is continuous at f(a). Given ε > 0, by the open mapping theorem,
f(B(a, ε)) is open. Since f(a) ∈ f(B(a, ε)), there is an open disk B(f(a), δ) ⊆ f(B(a, ε)) for some δ > 0.
Now w = f(f−1(w)). So,
|w− f(a)| < δ ⇒ w ∈ B(f(a), δ) ⊆ f(B(a, ε)) ⇒ f−1(w) ∈ B(a, ε)⇒ |f−1(w)− a| < ε.
Next we will show f ′(z) 6= 0 for all z ∈ U. Otherwise, suppose f ′(c) = 0. Then g(z) = f(z) − f(c)
has a root of order m ≥ 2 at c. By the local mapping theorem, there are distinct a and b near c such that
g(a) = g(b). Then f(a) = f(b), contradicting the fact f is injective.
Therefore, by the inverse rule, f−1 is holomorphic on f(U ). QED
Hurwitz’s Theorem. Let {fn} be a sequence of holomorphic functions on open Ω ⊆ C converging uniformly
on each closed disk in Ω to the (holomorphic, by Weierstrass’ theorem) function f . If fn has no roots in Ω
for all n, then either f has no roots in Ω or f ≡ 0. If fn is injective on Ω for all n, then either f is injective
on Ω or f is a constant function.
Proof. For the first assertion, suppose f 6≡ 0 and f has a root at w ∈ Ω. Then by the identity theorem,
there is a circle C = C(w, r) such that f(z) 6= 0 for z ∈ C and B(w, r) ⊆ Ω. Let ε = min
z∈C
|f(z)|, then ε > 0.
Since fn converges uniformly on B(w, r) to f , there is N such that n ≥ N ⇒ |fn(z) − f(z)| < ε for all
z ∈ B(w, r). Then for z ∈ C, |fn(z) − f(z)| < ε ≤ |f(z)| ≤ |fn(z)|+ |f(z)|. By Rouche´’s theorem, it follows
that fn must have a root inside C, which is a contradiction.
For the second assertion, suppose f(a) = f(b) for distinct a, b ∈ Ω and f is not the constant function
f(b). Then by the uniqueness theorem, there is a closed disk D centered at a in Ω such that f(z) 6= f(b) for
all z ∈ D \ {a}. In particular, b 6∈ D. Since fn converges uniformly on D to f , it follows that fn(z) − fn(b)
converges uniformly on D to f(z) − f(b). By the first assertion, since fn(z) − fn(b) has no roots in D,
f(z) − f(b) cannot have any root in D, which is a contradiction to f(a) = f(b). QED
Exercises
1. Find all possible values of I =
∫
C
dz
1 + z2
, where C is a curve with initial point 0 and final point 1 that
does not meet the poles of
1
1 + z2
.
2. Is there a holomorphic function f on the closed unit disk which sends the unit circle with the counter-
clockwise orientation into the unit circle with the clockwise orientation?
3. Determine the number of roots 17z5+ z2 + 1 has on the real and imaginary axis and in each quadrant.
4. Show that if α and β 6= 0 are real, the equation z2n + α2z2n−1 + β2 = 0 has n − 1 roots with positive
real parts if n is odd, and n roots with positive real parts if n is even.
5. If a > e, show that the equation ez = azn has n solutions inside the unit circle.
59
6. Suppose f is holomorphic and injective on B(a, r) = {z : |z − a| ≤ r}. If f has a root in B(a, r), show
that the root is given by
1
2pii
∫
|w−a|=r
wf ′(w)
f(w)
dw.
7. Suppose f is entire and f(z) is real if and only if z is real. Use the argument principle to show that f
can have at most one root. [Hint: Let Γ be a large circle |z| = R, what is n(f ◦ Γ, 0)?]
8. If f is holomorphic on and inside a simple closed curve Γ and f is injective on Γ, prove that f is injective
inside Γ. [Hint: Is f ◦ Γ a simple closed curve? For w 6∈ f ◦ Γ, let g(z) = f(z) − w, what is n(g ◦ Γ, 0)?]
9. Let P (z) be a polynomial of degree n ≥ 2.
(a) Show that
∫
|z|=r
dz
P (z)
= 0 when all the roots of P are inside the circle |z| = r.
(b) Suppose P has n distinct roots z1, . . . , zn. Show that
n∑
j=1
1
P ′(zj)
= 0.
10. Use Rouche´’s theorem to give another proof of the fundamental theorem of algebra.
11. Let f(z) = z +
∞∑
n=2
anz
n. Suppose
∞∑
n=2
n|an| ≤ 1.
(a) Prove that f is holomorphic on the open unit disk D, i.e. the power series converges for every
z ∈ D.
(b) Prove that f is injective on D. [Hint: Use Rouche´’s theorem to show that g(z) = f(z) − f(z0) has
exactly one solution in D for each fixed z0 ∈ D.]
12. Determine the number of roots z4 − 4z3 + 11z2 − 14z + 10 has on the real and imaginary axis and in
each quadrant.
13. Determine the number of roots z4 + 2z3 + 3z2 + z + 2 has on the real and imaginary axis and in each
quadrant.
14. Determine the number of roots 2iz2 + sin z has in {x+ iy : |x| ≤ pi/2, |y| ≤ 1}.
15. Prove that ez + z3 has no root in {z : |z| < 3/4} and has three roots in {z : |z| < 2}.
60
17. Applications of Residue Theory
Theorem. Let g, h be holomorphic near z0. If g has a root of order k at z0 (i.e. g(z0) = . . . = g(k−1)(z0) = 0,
g(k)(z0) 6= 0) and h has a root of order l at z0 (i.e. h(z0) = . . .= h(l−1)(z0) = 0, h(l)(z0) 6= 0), then g(z)
h(z)
has
a
{
removable singularity
pole (of order l − k)
}
at z0 in case
{
k ≥ l
k < l
}
.
Proof. Near z0,
g(z)
h(z)
=
∞∑
n=k
g(n)(z0)
n!
(z − z0)n
∞∑
n=l
h(n)(z0)
n!
(z − z0)n
=
g(k)(z0)
k!
l!
h(l)(z0)
(z − z0)k−l + . . .. If k − l ≥ 0, then z0 is a
removable singularity. If k − l < 0, then it is a pole of order l − k. QED
If f is holomorphic at z0, Res(f, z0) = 0 (because an = 0 for n < 0). For poles, we have the
ϕ-Method. If f has a pole of order k at z0, then the Laurent expansion of f(z) is
a−k
(z − z0)k +. . .+
a−1
(z − z0) +
a0+a1(z−z0)+. . .. So the function ϕ(z) = (z−z0)kf(z) = a−k+. . .+a−1(z−z0)k−1+a0(z−z0)k+. . . is holo-
morphic at z0. Then Res(f, z0) = a−1 =
ϕ(k−1)(z0)
(k − 1)! = limz→z0
ϕ(k−1)(z)
(k − 1)! = limz→z0
1
(k − 1)!
dk−1
dzk−1
[(z − z0)kf(z)].
Special Cases: (1) If f =
g
h
, z0 is a root of order k for g, a root of order k + 1 for h, then z0 is a simple
pole (i.e. pole of order 1) and Res(f, z0) = (k + 1)
g(k)(z0)
h(k+1)(z0)
.
(2) If f =
g
h
, g(z0) 6= 0, h(z0) = h′(z0) = 0, h′′(z0) 6= 0, then z0 is a double pole (i.e. pole of order 2)
and Res(f, z0) = 2
g′(z0)
h′′(z0)
− 2
3
g(z0)h′′′(z0)
(h′′(z0))2
. If f =
g
h
, g(z0) = 0, g′(z0) 6= 0, h(z0) = h′(z0) = h′′(z0) = 0,
h′′′(z0) 6= 0, then z0 is a double pole and Res(f, z0) = 3 g
′′(z0)
h′′′(z0)
− 2
3
g′(z0)h(4)(z0)
(h′′′(z0))2
.
Examples. (1)
z
1 + ez
has a pole at pii. It is a simple pole. So,
Res(
z
1 + ez
, pii) = lim
z→pii
(z − pii) z
1 + ez
=
pii
epii
= −pii.
(2)
1
z sin z
has a pole at 0. It is a double pole. So,
Res(
1
z sin z
, 0) = lim
z→0
1
1!
(
z2
1
z sin z
)′
= lim
z→0
sin z − z cos z
sin2 z
= lim
z→0
z
2 cos z
= 0.
Alternatively, near 0,
1
z sin z
=
1
z(z − 16z3 + · · ·)
=
1
z2 − 16z4 + · · ·
=
1
z2
+
1
6
+ · · · . So, Res( 1
z sin z
, 0) = 0.
(3)
cot z
z(z − 1) =
cos z
z(z − 1) sin z has a pole at 0. It is a double pole. Let g(z) = cos z, h(z) = z(z−1) sin z, then
g(0) = 1, g′(0) = 0, h′′(0) = −2, and h′′′(0) = 6. So, Res( cot z
z(z − 1) , 0) = 2
0
−2 −
2
3
1× 6
(−2)2 = −1.
61
Alternatively, near 0,
cos z
z(z − 1) sin z =
1− z22 + · · ·
(−z + z2) (z − z36 + · · ·) =
1− z22 + · · ·
−z2 + z3 + · · · = −
1
z2
− 1
z
+ · · · .
So, Res(
cot z
z(z − 1) , 0) = −1.
(4) z cos
1
z + 1
has an essential singularity at −1, since near −1,
z cos
1
z + 1
= [(z + 1)− 1]
[
1− 1
2(z + 1)2
+ · · ·
]
= (z + 1)− 1− 1
2(z + 1)
+ · · · .
So Res(z cos
1
z + 1
,−1) = −1
2
.
Recall the residue theorem implies that if f is holomorphic on and inside a simple closed curve C except
at poles a1, . . . , an, then
∫
C
f(z) dz = 2pii
n∑
j=1
Res(f, aj). (In view of the examples above, notice this formula
said something amazing, namely the integral on a closed curve can be computed by doing some derivatives!)
Now let us compute some nonelementary integrals.
Some Common Types of Integrals.
Type I.
∫ 2pi
0
F (sin kθ, cosmθ) dθ =
∫
|z|=1
F
(
zk − z−k
2i
,
zm + z−m
2
)
dz
iz
, where z = eiθ, dz = ieiθ dθ.
Example.
∫ 2pi
0
dθ
5 + 4 sin θ
=
∫
|z|=1
1
5 + 4
(
z − z−1
2i
) dz
iz
=
∫
|z|=1
dz
(2z + i)(z + 2i)
= 2pii Res
z=− i2
1
(2z + i)(z + 2i)
=
2
3
pi, where we observed that only − i
2
is inside |z| = 1 and it is a simple pole.
Type II.
∫ ∞
−∞
P (x)
Q(x)
dx, where P , Q are polynomials and Q has no real roots.
Remark. The improper integral
∫ ∞
−∞
is defined (in the Riemann sense) to be lim
a→−∞
∫ 0
a
+ lim
b→+∞
∫ b
0
. If
∫ ∞
−∞
exists, it agrees with the limit lim
R→+∞
∫ R
−R
(which is called the principal value of
∫ ∞
−∞
and is denoted by P.V.∫ ∞
−∞
). If
∫ ∞
−∞
doesn’t exist, sometimes P.V.
∫ ∞
−∞
may exist.
Example.
∫ ∞
−∞
x dx = lim
a→−∞
∫ 0
a
x dx+ lim
b→+∞
∫ b
0
x dx doesn’t exist, but P.V.
∫ ∞
−∞
x dx = 0.
If degQ(x) > degP (x) + 1, then
∫ ∞
−∞
P (x)
Q(x)
dx exists (because near ±∞, it is like
∫ ±∞
c
1
xn
dx, n > 1).
In this case, we often compute its principal value instead. If degQ(x) ≤ degP (x) + 1, then
∫ ∞
−∞
P (x)
Q(x)
dx
doesn’t exist, but P.V.
∫ ∞
−∞
P (x)
Q(x)
dx may exist.
62
Example. Find
∫ ∞
−∞
x2 − 1
(x2 + 1)(x2 − 2x+ 2) dx
(
= lim
R→∞
∫ R
−R
f(x) dx
)
.
Factoring the denominator, we get (x2 + 1)(x2 − 2x+ 2) = (x− i)(x + i)(x − 1− i)(x− 1 + i).
Step 1 . Consider the contour
CR
R0-R
1 -i-i
+i1i
where R is large.
By the residue theorem,∫
CR+[−R,R]
f(z) dz = 2pii
(
Res
z=i
f(z) + Res
z=1+i
f(z)
)
= 2pii
(
lim
z→i
z2 − 1
(z + i)(z2 − 2z + 2) + limz→1+i
z2 − 1
(z2 + 1)(z − 1− i)
)
=
pi
5
.
Step 2 . Now z ∈ CR ⇒ |z| = R,
∣∣∣∣∫
CR
f(z) dz
∣∣∣∣ ≤ML = R2 + 1(R2 − 1)(R2 − 2R− 2)piR→ 0 as R→ +∞.
Step 3 . So,
∫ ∞
−∞
f(x) dx = lim
R→∞
∫ R
−R
f(x) dx = lim
R→∞
( ∫
CR+[−R,R]
f(z) dz −
∫
CR
f(z) dz
)
=
pi
5
.
Type III.
∫ ∞
−∞
P (x)
Q(x)
{
cos ax
sin ax
}
dx =
{
Re
Im
}∫ ∞
−∞
P (x)
Q(x)
eiax dx, where P , Q are polynomials and Q has no
real roots.
If degQ(x) > degP (x) + 1, then
∫ ∞
−∞
P (x)
Q(x)
{
cos ax
sin ax
}
dx exists (and equals its principal value). If
degQ(x) ≤ degP (x) + 1, then
∫ ∞
−∞
P (x)
Q(x)
{
cos ax
sin ax
}
dx or its principal value may exist.
The following is a useful inequality when dealing with Type III integrals.
Jordan’s Inequality. For M > 0,
∫ pi
0
e−M sin θ dθ = 2
∫ pi
2
0
e−M sin θ dθ ≤ 2
∫ pi
2
0
e−M
2θ
pi dθ <
pi
M
(because on
[0,
pi
2
], sin θ ≥ 2θ
pi
as can be seen from the graphs of y = sin θ and y =
2θ
pi
).
Example. Find
∫ ∞
−∞
x sin 9x
x2 + 4
dx
(
= Im
∫ ∞
−∞
zei9z
z2 + 4
dz
)
. (Note
x sin 9x
x2 + 4
is even. If the principal value of
the integral exists, then
∫ 0
−∞
=
1
2
P.V.
∫ ∞
−∞
=
∫ ∞
0
and the integral (in the Riemann sense) will exist.)
Step 1 . Consider the contour
CR
R0-R
i2
-2 i
where R is large.
By the residue theorem,
∫
CR+[−R,R]
zei9z
z2 + 4
dz = 2piiRes
z=2i
zei9z
z2 + 4
= 2pii lim
z→2i
zei9z
z + 2i
= e−18pii.
63
Step 2 . As R→∞,∣∣∣∣∫
CR
zei9z
z2 + 4
dz
∣∣∣∣ ≤ ∫
CR
∣∣∣∣ zz2 + 4
∣∣∣∣ ∣∣ei9z∣∣ |dz| ≤ RR2 − 4
∫ pi
0
e−9R sin θRdθ ≤ R
R2 − 4R
pi
9R
→ 0,
where we have applied Jordan’s inequality to get the last inequality.
Step 3 . Therefore,
P.V.
∫ ∞
−∞
x sin 9x
x2 + 4
dx = Im
(
lim
R→∞
( ∫
CR+[−R,R]
zei9z
z2 + 4
dz −
∫
CR
zei9z
z2 + 4
dz
))
= e−18pi.
We extract from step 2 a useful fact.
Jordan’s Lemma. If lim
R→∞
(max
z∈CR
|h(z)|) = 0, then lim
R→∞
∫
CR
h(z)eiaz dz = 0 for a > 0.
Proof. As R→∞,∣∣∣∣∫
CR
h(z)eiaz dz
∣∣∣∣ ≤ ∫
CR
|h(z)| ∣∣eiaz∣∣ |dz| ≤ max
z∈CR
|h(z)|
∫ pi
0
e−aR sin θRdθ ≤ pi
a
max
z∈CR
|h(z)| → 0.
QED
Big O-Little o Notations. We say f(z) = O(g(z)) as z → a if there is M such that
∣∣∣∣f(z)g(z)
∣∣∣∣ ≤M as z → a
(i.e. in a neighborhood of a). We say f(z) = o(g(z)) as z → a if f(z)
g(z)
→ 0 as z → a.
Type IV.
∫ ∞
0
xαf(x) dx (−1 < α < 0) or
∫ ∞
0
(lnx)kf(x) dx (k = 1, 2, 3 . . .), where f is meromorphic on C
and continuous at 0. (In the first integral, we require f(z) = o(
1
|z|α+1 ) as z →∞ and in the second integral,
we require f(z) = O(
1
|z|2 ) as z →∞. These conditions ensure the integrals will exist.)
Example. Find
∫ ∞
0
dx√
x(x2 + 1)
= lim
R→∞
r→0+
∫ R
r
x−
1
2
1
x2 + 1
dx.
Step 1 . Consider the contour
−γ 2
γ 1
r
i
-i
R
θ
CR
-C r
where r is small and R is large.
Here γ1(x) = x, γ2(x) = xei(2pi−θ), r ≤ x ≤ R. Inside and on the contour, we have 0 ≤ arg z < 2pi.
By the residue theorem,
∫
CR−γ2−Cr+γ1
z−
1
2
1
z2 + 1
dz = 2pii
(
Res
z=i
z−1/2
z2 + 1
+ Res
z=−i
z−1/2
z2 + 1
)
= 2pii
(
lim
z→i
z−1/2
z + i
+ lim
z→−i
z−1/2
z − i
)
= pi(i−1/2 − (−i)−1/2) = pi(e−ipi/4 − e−3ipi/4) = pi√2.
64
Step 2 . We have
∫
γ1
z−1/2
z2 + 1
dz =
∫ R
r
x−1/2
x2 + 1
dx, lim
θ→0+
∫
γ2
z−1/2
z2 + 1
dz =
∫ R
r
x−1/2e−ipi
x2 + 1
dx = −
∫ R
r
x−1/2
x2 + 1
dx,∣∣∣∣∫
CR
z−1/2
z2 + 1
dz
∣∣∣∣ ≤ML = R−1/2R2 − 1(2pi − θ)R→ 0 as R→ +∞ and∣∣∣∣∫
Cr
z−1/2
z2 + 1
dz
∣∣∣∣ ≤ML = r−1/21− r2 (2pi − θ)r → 0 as r → 0+.
Step 3 . We have pi
√
2 = lim
r,θ→0+,R→+∞
∫
CR−γ2−Cr+γ1
z−1/2
1
z2 + 1
dz = 2
∫ ∞
0
x−1/2
x2 + 1
dx.
Therefore,
∫ ∞
0
dx√
x(x2 + 1)
=
pi
√
2
2
.
Miscellaneous Examples
Example. Find
∫ ∞
−∞
sinx
x
dx =
(
Im
∫ ∞
−∞
eix
x
dx
)
. (Note
sinx
x
is even.)
Step 1 . Consider the contour
R-R -r r
-C r
CR
where r is small and R is large.
By the residue theorem,
(∫
CR
+
∫ −r
−R
−
∫
Cr
+
∫ R
r
)
eiz
z
dz = 0 (since there is no singularity inside the con-
tour).
Step 2 . Since lim
R→+∞
max
z∈CR
∣∣∣∣1z
∣∣∣∣ = limR→+∞ 1R = 0, Jordan’s lemma implies limR→+∞
∫
CR
eiz
z
dz = 0. Now
∫
Cr
eiz
z
dz =
∫
Cr
(
1 + iz +
(iz)2
2
+ . . .
)
z
dz =
∫
Cr
(
1
z
+ g(z)
)
dz = pii+
∫
Cr
g(z) dz,
where g(z) has a removable singularity (hence holomorphic) at 0, so it is bounded near 0 (say by M ). Then,∣∣∣∣∫
Cr
g(z) dz
∣∣∣∣ ≤ML =Mpir → 0 as r → 0+.
Step 3 . Therefore,
0 = lim
r→0+,R→+∞
(∫
CR
+
∫ −r
−R
−
∫
Cr
+
∫ R
r
)
eiz
z
dz =
∫ ∞
−∞
eix
x
dx− pii⇒
∫ ∞
−∞
sinx
x
dx = Im
∫ ∞
−∞
eix
x
dx = pi.
We extract from step 2 the following useful rule for dealing with arcs around simple poles.
Rule. If f(z) has an isolated simple pole at z = c and Cr is the arc z(θ) = c + reiθ, α ≤ θ ≤ β, then
lim
r→0+
∫
Cr
f(z) dz = i(β − α)Res
z=c
f(z).
65
β
α
C r
c
Proof. The Laurent expansion of f at c is
a−1
z − c +
∞∑
k=0
ak(z − c)k = a−1
z − c + g(z) where
g(z) is holomorphic at c. So∫
Cr
f(z) dz =
∫
Cr
(
a−1
z − c + g(z)
)
dz = a−1
∫ β
α
1
reiθ
ireiθ dθ +
∫
Cr
g(z) dz
= i(β − α)Res
z=c
f(z) +
∫
Cr
g(z) dz.
Now g is bounded near c (say by M ), then
∣∣∣∣∫
Cr
g(z) dz
∣∣∣∣ ≤ML = M (β − α)r→ 0 as r→ 0+. QED
Example. Find
∫ ∞
−∞
eax
1 + ex
dx, where 0 < a < 1. (Note the integrand is positive. For R > r > 0,∫ r
−r
≤
∫ 0
−r
+
∫ R
0
,
∫ 0
−R
+
∫ r
0
≤
∫ R
−R
so that P.V.
∫ ∞
−∞
=
∫ ∞
−∞
.)
Step 1 . Consider the function f(z) =
eaz
1 + ez
, its poles are at npii (n odd integer).
3pi i
pi i
−pi i
γ1
−γ 2
−γ 3
-R R
Consider the contour on the left, where R is large.
γ1(t) = R+ it, 0 ≤ t ≤ 2pi,
γ2(t) = t+ 2pii,−R ≤ t ≤ R,
γ3(t) = −R+ it, 0 ≤ t ≤ 2pi.
By the residue theorem,
(∫ R
−R
+
∫
γ1−γ2−γ3
)
f(z) dz = 2pii Res
z=pii
f(z) = 2pii lim
z→pii
(z − pii)eaz
1 + ez
= −2piiepiai.
Step 2 . We have
∣∣∣∣∫
γ1
eaz
1 + ez
dz
∣∣∣∣ = ∣∣∣∣∫ 2pi
0
ea(R+it)i dt
1 + eR+it
∣∣∣∣ ≤ML = eaReR − 12pi→ 0 as R→ +∞ (because a < 1),∫
γ2
eaz
1 + ez
dz =
∫ R
−R
ea(t+2pii)
1 + et+2pii
dt = e2piai
∫ R
−R
eat
1 + et
dt and∣∣∣∣∫
γ3
eaz
1 + ez
dz
∣∣∣∣ = ∣∣∣∣∫ 2pi
0
ea(−R+it)i dt
1 + e−R+it
∣∣∣∣ ≤ML = e−aR1− e−R 2pi → 0 as R→ +∞ (because 0 < a).
Step 3 . So, −2piiepiai = lim
R→+∞
(∫ R
−R
+
∫
γ1−γ2−γ3
)
f(z) dz = (1− e2piai)
∫ ∞
−∞
eax
1 + ex
dx.
Therefore,
∫ ∞
−∞
eax
1 + ex
dx =
−2piiepiai
1− e2piai = pi
(
2i
epiai − e−piai
)
=
pi
sinpia
.
The following example illustrates how the residue theorem can produce interesting series representations
of some common functions.
Example. The poles of cot z =
cos z
sin z
are at kpi, k any integer. For a fixed z 6= kpi, let n be large so that
z is inside the square Γn with vertices ±(n + 12)pi ± i(n+
1
2
)pi. Now for w = (n + 12 )pi + iy on the right
edge of Γn, | cotw| =
∣∣∣∣eiw + e−iweiw − e−iw
∣∣∣∣ = ∣∣∣∣e−y − eye−y + ey
∣∣∣∣ ≤ 1 and for w = x + i(n + 12 )pi on the top edge of Γn,
66
| cotw| =
∣∣∣∣e2iw + 1e2iw − 1
∣∣∣∣ ≤ 1 + e−(2n+1)pi1− e−(2n+1)pi ≤ 1 + e−pi1− e−pi = C. Since | cotw| = | cot(−w)| and 1 ≤ C, so | cotw| ≤ C
on Γn. By the M -L inequality, as n→∞,∣∣∣∣∫
Γn
z cotwdw
w(w − z)
∣∣∣∣ ≤ |z|C(n+ 12)pi ((n + 12 )pi − |z|)(8n+ 4)pi→ 0.
By the residue theorem,
0 = lim
n→∞
1
2pii
∫
Γn
z cotwdw
w(w − z) = limn→∞
(
Res
w=z
z cotw
w(w − z) +
n∑
k=−n
Res
w=kpi
z cotw
w(w − z)
)
= cot z − 1
z
+
∑
k 6=0
z
kpi(kpi − z) = cot z −
1
z
−
∞∑
k=1
2z
z2 − k2pi2
(where the double pole at 0 has residue −1
z
due to
z cosw
w(w − z) sinw =
z − zw2/6 + · · ·
−zw2 + w3 + · · · = −
1
w2
− 1
zw
+ · · ·
by long division.) Therefore, cot z =
1
z
+
∞∑
k=1
2z
z2 − k2pi2 =
1
z
+
∞∑
k=1
(
1
z − kpi +
1
z + kpi
)
for z 6= kpi, k ∈ Z .
Remarks. The last example can be modified to yield formulas for summing series. To be more precise, let
f be meromorphic on C with finitely many poles zp, p = 1, 2, . . . , k and lim
z→∞ zf(z) = 0. Suppose Cn is the
counterclockwise square with vertices ±(n+ 1
2
)± i(n + 1
2
), then by the residue theorem,
∫
Cn
pif(z) cot piz dz = 2pii
 n∑
j=−n
j 6=zp
f(j) +
k∑
p=1
Res (pif(z) cot piz, zp)
 .
Since lim
z→∞ zf(z) = 0 and n+
1
2
≤ |z| ≤ (n + 1
2
)
√
2 for all z ∈ Cn, we have lim
n→∞(n+
1
2
) max
z∈Cn
|f(z)| = 0. By
the M -L inequality, as n→∞,∣∣∣∣∫
Cn
pif(z) cot piz dz
∣∣∣∣ ≤ pi(maxz∈Cn |f(z)|
)
1 + e−pi
1− e−pi (8n+ 4)→ 0.
It follows that
∞∑
j=−∞
j 6=zp
f(j) = −
k∑
p=1
Res (pif(z) cot piz, zp) .
For instance, if we set f(z) =
1
z2
, then
∞∑
j=1
1
j2
=
1
2
∞∑
j=−∞
j 6=0
1
j2
= −1
2
Res
(
pi cotpiz
z2
, 0
)
=
pi2
6
.
If we replace cot z by csc z in the above argument, then we get a similar formula
∞∑
j=−∞
j 6=zp
(−1)jf(j) = −
k∑
p=1
Res (pif(z) csc piz, zp) .
67
Exercises
1. Show that
∫ pi
2
0
dθ
1 + sin2 θ
=
pi
2
√
2
and
1
2pi
∫ 2pi
0
(2 cos θ)2n dθ =
(2n)!
n!n!
for every positive integer n.
2. Find
∫ ∞
0
dx
1 + xn
, where n ≥ 2 is a positive integer. [Hint: This can be done following the example for
Type II integrals. Alternatively, the contour below can be considered.]
Re
0 R
2pi i/n
3. Find
∫ ∞
−∞
sin2 x
x2
dx,
∫ ∞
−∞
sin3 x
x3
dx and
∫ ∞
−∞
sin2 x
1 + x2
dx. [Hint: 4 sin3 x = Im(3eix − e3ix).]
4. Given
∫ ∞
0
e−x
2
dx =
√
pi
2
. Find
∫ ∞
0
cos x2 dx and
∫ ∞
0
sinx2 dx. [ Hint: Use the contour
0 R
pi/4 . ]
5. Find
∫ ∞
0
lnx
x2 + 1
dx. [ Hint: Use the contour
-r r R-R
. ]
6. Given that
∫ ∞
0
e−x
2
dx =
√
pi
2
. Find
∫ ∞
0
e−x
2
cos 2x dx.
[Hint: Use the contour
R
R+i
-R
-R+i
and f(z) = e−z
2
. ]
7. Suppose f is holomorphic on the annulus {z : r < |z| <∞}. Define Res(f,∞) = − 1
2pii
∫
|z|=R>r
f(z) dz.
oo
0
(Here, the minus sign appears because the counterclockwise orientation around ∞ cor-
responds to the clockwise orientation around 0 as can be seen from stereographic projec-
tion.) Equivalently, if f(z) =
∞∑
k=−∞
akz
k on {z : r < |z| <∞}, then Res(f,∞) = −a−1.
(a) If f is meromorphic on C with isolated poles at a1, . . . , an, show that
n∑
j=1
Res(f, aj)+Res(f,∞) = 0.
That is, the sum of all residues in C∪{∞} is 0.
(b) Show that Res(f(z),∞) = Res
(
− 1
z2
f
(
1
z
)
, 0
)
.
(c) Find
∫
|z|=1
dz
sin
(
1
z
) .
8. Find
∞∑
j=1
1
j4
,
∞∑
j=1
(−1)j+1
j2
and
∞∑
j=1
1
1 + j2
using the residue theorem.
68
18. Infinite Products
Suppose a polynomial P (z) has roots at 0 (of order m), z1, z2, . . . , zn. Then
P (z) = Azm
n∏
k=1
(z − zk) = Bzm
n∏
k=1
(
1− z
zk
)
,
where B = A× (−z1) × · · · × (−zn) and can be determined from P , e.g. B = lim
z→0
P (z)
zm
.
Now sin z is entire and has simple roots at kpi, k any integer. If we think of (the power series of) an
entire function as an “infinite” polynomial, then we may conjecture that
sin z = z
∞∏
k=1
(
1− z
kpi
)(
1 +
z
kpi
)
= z
∞∏
k=1
(
1− z
2
k2pi2
)
.
This formula is true for all complex numbers and was first discovered by Euler. Taking z =
pi
2
and
transposing terms, we get
pi
2
=
∞∏
k=1
4k2
4k2 − 1 =
(
2× 2
1× 3
)(
4× 4
3× 5
)(
6× 6
5× 7
)(
8× 8
7× 9
)
· · · ,
which is known as Wallis’ Formula. Also we have
z − z
3
6
+ · · ·= sin z = z
∞∏
k=1
(
1− z
2
k2pi2
)
= z
[
1−
( ∞∑
k=1
1
k2pi2
)
z2 + · · ·
]
.
By the uniqueness of the coefficients of z3, we deduce that
∞∑
k=1
1
k2
=
pi2
6
.
Now to prove Euler’s formula for sin z, we begin with the definition of infinite products.
Definitions. For a sequence {ak} of nonzero complex numbers, we define
∞∏
k=1
ak = lim
n→∞
n∏
k=1
ak. If the limit
is a nonzero number, we say
∞∏
k=1
ak converges. Otherwise, we say
∞∏
k=1
ak diverges. For sequences with finitely
many of the ak’s being 0 and
∞∏
k=1,ak 6=0
ak converges, we also say
∞∏
k=1
ak converges to 0.
Examples. (1)
∞∏
k=2
(
1− 1
k
)
= lim
n→∞
n∏
k=2
k − 1
k
= lim
n→∞
1
n
diverges to 0.
(2)
∞∏
k=1
(
1 +
1
k
)
= lim
n→∞
n∏
k=1
k + 1
k
= lim
n→∞(n + 1) diverges to ∞.
69
(3)
∞∏
k=2
(
1− 1
k2
)
= lim
n→∞
n∏
k=2
(k − 1)(k + 1)
k2
= lim
n→∞
n + 1
2n
=
1
2
. (Note that 1− 1
k2
=
(
1 +
1
k
)(
1− 1
k
)
. If
we consider the factor 1− 1
k
as a “weight”, then we see that an infinite product can be made to converge
by putting proper weights on the terms.)
Term Test. If
∞∏
k=1
ak converges, then
lim
k→∞
ak = lim
k→∞
 k∏
p=1,ap 6=0
ap
/ k−1∏
p=1,ap 6=0
ap
 = ∞∏
p=1,ap 6=0
ap
/ ∞∏
p=1,ap 6=0
ap = 1.
Remark. For convergent products, the terms tend to 1. Since the logarithm of a finite product is the sum
of the logarithms, we see easily that
∞∏
k=1
ak converges if and only if
∞∑
k=1,ak 6=0
Log ak converges.
In the lemma below, it will be shown that for |z| ≤ 1
2
, |Log(1−z)| ≤ 2|z|. So, by the absolute convergence
test and the comparison test,
∞∑
k=1
|1− ak| converges implies
∞∏
k=1
ak converges.
Definition. An elementary factor is one of the following entire functions
E0(z) = 1− z and Ep(z) = (1− z)ez +
z2
2 + · · ·+ z
p
p for p = 1, 2, 3, . . ..
Lemma. If |z| ≤ 1
2
, then |LogEp(z)| ≤ 2|z|p+1.
Proof. For |z| ≤ 1
2
, let C be the line segment from 0 to z. Then
Log(1− z) = −
∫
C
dw
1− w = −
∫
C
∞∑
k=0
wk dw = −
∞∑
k=0
zk+1
k + 1
and
|LogEp(z)| =
∣∣∣∣∣∣
∞∑
k=p
zk+1
k + 1
∣∣∣∣∣∣ ≤
∞∑
k=p
|z|k+1
k + 1
≤ |z|
p+1
p+ 1
∞∑
k=p
1
2k−p
≤ 2|z|p+1.
QED
Theorem. Let {zk} be a sequence of nonzero complex numbers going to ∞ and {mk} a sequence of non-
negative integers such that for any fixed R > 0,
∞∑
k=1
(
R
|zk|
)mk
converges (e.g. mk = k because
R
|zk| <
1
2
for
k large). Then the product
∞∏
k=1
Emk−1
(
z
zk
)
=
∞∏
k=1
(
1− z
zk
)
e
z
zk
+ z
2
2z2
k
+ · · ·+ zmk−1
(mk−1)zmk−1k
converges to an entire function with roots zk (repeated according to multiplicities.)
70
Proof. Fix R > 0. As |zk| → ∞, there exists N ∈ N be such that k > N ⇒ 2R < |zk|. Then for |z| ≤ R and
k > N , we have
∣∣∣∣ zzk
∣∣∣∣ < 12 . By the lemma,
∞∑
k=N+1
∣∣∣∣LogEmk−1( zzk
)∣∣∣∣ ≤ 2 ∞∑
k=N+1
(
R
|zk|
)mk
<∞.
By Weierstrass’ M -test (and Weierstrass’ theorem in chapter 14),
∞∑
k=N+1
LogEmk−1
(
z
zk
)
converges uni-
formly (to a holomorphic function) on |z| ≤ R. Exponentiating and multiplying the first N terms, we see
that
∞∏
k=1
Emk−1
(
z
zk
)
is holomorphic on |z| < R. Since R is arbitrary,
∞∏
k=1
Emk−1
(
z
zk
)
must be entire.
Clearly, the zk’s are roots. For w 6= zk, Emk−1
(
w
zk
)
6= 0 and the convergence of
∞∏
k=1
Emk−1
(
w
zk
)
implies lim
K→∞
∞∏
k=K+1
Emk−1
(
w
zk
)
= 1. So for K large,
∞∏
k=K+1
Emk−1
(
w
zk
)
6= 0. Then
∞∏
k=1
Emk−1
(
w
zk
)
=
K∏
k=1
Emk−1
(
w
zk
) ∞∏
k=K+1
Emk−1
(
w
zk
)
6= 0. Therefore, there cannot be any other roots. QED
Weierstrass Factorization Theorem. Let f be an entire function with roots 0 (of order m ≥ 0), z1, z2, . . .
going to ∞. Suppose there is a sequence {mk} of nonnegative integers as in the previous theorem, then
there is an entire function g such that
f(z) ≡ eg(z)zm
∞∏
k=1
(
1− z
zk
)
e
z
zk
+ · · ·+ zmk−1
(mk−1)zmk−1k .
Proof. By the previous theorem, h(z) = zm
∞∏
k=1
Emk−1
(
z
zk
)
is entire and has the same roots (with the
same multiplicities) as f(z). Then
f
h
is an entire function without any roots. By the logarithm theorem in
chapter 10, f/h = eg for some entire function g. Therefore, f = egh. QED
Corollary. Every meromorphic function g on C is equal to f1/f0 for some holomorphic functions f0 and f1
on C .
Proof. By the theorem, there exists a holomorphic function f0 on C having roots of the same multiplicities
as the order of poles at the poles of g. Then f1 = gf0 is holomorphic on C and g = f1/f0. QED
The corollary is also true for open sets in C, see Robert Greene and Steven Krantz’s book Function Theory
of One Complex Variable, 3rd edition, p. 270.
Now Euler’s formula for sin z follows easily. Since
∞∑
k=1
((
R
kpi
)2
+
(
R
−kpi
)2)
<∞ for all R > 0, we
may take mk = 2 for all k. By the Weierstrass factorization theorem,
sin z = eg(z)z
∞∏
k=1
(
1− z
kpi
)
ez/kpi
(
1 +
z
kpi
)
e−z/kpi = eg(z)z
∞∏
k=1
(
1− z
2
k2pi2
)
.
71
Taking logarithmic derivatives, cot z =
d
dz
(log sin z) = g′(z) +
1
z
+
∞∑
k=1
2z
z2 − k2pi2 . By the last example of
chapter 17, g is constant. By the lemma, for all z ∈ B(0, 1/2),
∞∑
k=1
Log
(
1− z
2
k2pi2
)
≤ 2
∞∑
k=1
1
k2pi2
<∞. By
M -test, uniform convergence follows. Hence lim
z→0
∞∑
k=1
Log
(
1− z
2
k2pi2
)
= 0 and eg = lim
z→0
sin z
z
∞∏
k=1
(
1− z
2
k2pi2
)−1
= 1. Therefore,
sin z = z
∞∏
k=1
(
1− z
2
k2pi2
)
.
Exercises
1. Prove that
(a)
∞∏
n=1
(
1 +
1
n(n + 2)
)
= 2; (b)
∞∏
n=2
n3 − 1
n3 + 1
=
2
3
.
2. Prove that
∞∏
n=1
cos
α
2n
=
sinα
α
and
2√
2
2√
2 +
√
2
2√
2 +
√
2 +
√
2
· · · = pi
2
.
3. Find
∞∏
n=2
(
1 +
(−1)n−1
an
)
, where an =
n−1∑
k=1
(−1)k−1n!
k!
.
4. If |z| < 1, prove that
∞∏
n=0
(1 + z2
n
) =
1
1− z .
5. Suppose
∞∑
n=1
|zn|2 <∞. Is it necessary that
∞∏
n=1
cos zn must converge?
6. Show that
∞∑
k=1
(1− ak) converges does not imply
∞∏
k=1
ak converges by considering ak = 1 +
(−1)k−1√
k
.
72
Suggested Readings and References
There are a large number of textbooks on complex analysis. Some good texts for reference alongside
this set of notes are:
[1] J. Bak and D. J. Newman, Complex Analysis, Springer-Verlag, 1982;
[2] R. P. Boas, Invitation to Complex Analysis, Random House, 1987;
[3] J. E. Marsden and M. J. Hoffman, Basic Complex Analysis, 2nd ed.,Freeman, 1987;
[4] Yu. V. Sidorov, M. V. Fedoryuk and M. I. Shabunin, Lectures on the Theory of Functions of a Complex
Variable, Mir, 1985.
For students who like to have a simple glance at applications in science and engineering, we refer to
[5] S. Fisher, Complex Variables, 2nd ed., Brooks/Cole, 1990.
For mature students who like to read more about the subject, there are the classics:
[6] L. Ahlfors, Complex Analysis, 3rd ed., Mc Graw-Hill, 1979;
[7] S. Saks and A. Zygmund, Analytic Functions, 3rd ed., Elsevier, 1971;
[8] E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford, 1939.
For students who like to do more problems, we recommend
[9] K. Knopp, Problem Book in the Theory of Functions, vol. I & II, Dover, 1948,1952;
[10] J. G. Krzyz, Problems in Complex Variable Theory, Elsevier, 1971;
[11] L. I. Volkovyskii, G. L. Lunts and I. G. Aramanovich, A Collection of Problems on Complex Analysis,
Dover, 1991.
In passing, we will like to mention that there are many well-written historical accounts of the subject
appeared in the periodical The Mathematical Intelligencer, published by Springer-Verlag. On the other hand,
for those who prefer to read a book on the development of the subject with some history, the following is
highly recommended:
[12] R. Remmert, Theory of Complex Functions, Springer-Verlag, 1991.
Among the references above, [1] and [4] are the best to read along with this set of notes. One of the
many outstanding features of [4] is its large collection of examples, from simple to advanced. This reference
also have excellent discussions on conformal mappings, multiple-valued analytic functions and asymptotic
methods. On the other hand, [1] is a combination of beauty and elegance. There are interesting materials in
[1] which do not appear anywhere else in the literature. Also, in [1], there are many ingenious ideas useful
in different parts of mathematics and clever proofs of important theorems. For [2] and [3], the students
will find interesting techniques, useful formulas and detailed discussions on specific materials. For example,
the discussions on analytic continuation in [2] is very, very well presented, informative and worthy of any
student’s attention.
As for exercises, the problems in [9], [10] and [11] are very good for practice. Some are quite challenging
and most will serve to strengthen the students’ understanding.
For students who like to continue pursuing the subject, [8] should be a good place to start, then the
second half of [6] and most of [7] would provide a solid foundation. Finally, those who plan to eventually
specialize in analysis, we suggest the standard graduate texts:
[13] J. B. Conway, Functions of One Complex Variable, 2nd ed., Springer-Verlag, 1978;
[14] W. Rudin, Real and Complex Analysis, 3rd ed., Mc Graw-Hill, 1987.
73
Index
absolute convergence test 2
analytic 11
annulus 20, 51
argument principle 57
big O notation 64
bilinear transformation 23
boundary 4
bounded set 4
branch 19
Casaroti-Weierstrass theorem 53
Cauchy integral formula 36, 40, 56
for derivatives 40, 56
Cauchy sequence 1
Cauchy-Riemann equations 16, 18
polar form 18
Cauchy-Schwarz inequality 3
Cauchy’s theorem 32, 34
for homotopic curves 34
on disks 32
chordal metric 11
closed curve 29
closed curve theorem 30
closed disk 4
closed set 4
closure 4
compact 4
conformal mapping 22
conformally equivalent 22
connected 4
continuous 7, 11
contour integral 29
converge 1, 2, 51, 69
infinite product 69
sequence 1
series 2, 51
converge uniformly 7
cross ratio 24
differentiable 11
disconnected 4
disk of convergence 13
diverge 1, 2, 69
infinite prouct 69
sequence 1
series 2
domain 4
double root 41
elementary factor 70
entire 11
essential singularity 53, 54
Euler’s equation 19
exponential function 19, 20
extended complex plane 10
extension theorem 33
fixed point 23, 44
fundamental theorem 30, 41
of algebra 41
of calculus 30
Goursat’s theorem 32
great Picard theorem 53
harmonic 46
conjugate 46
holomorphic 11
holomorphic part 53
Hurwitz’s theorem 59
identity theorem 14, 41
for power series 14
improper integral 62
principal value 62
Riemann sense 62
infinite product 69
integral function 11
integral of contour 29
integral theorem 32, 34
interior 4
inverse mapping theorem 59
inverse of function 19
inverse rule 20
isolated singulaity 52
Jordan’s inequality 63
Jordan’s lemma 64
L’ Hoˆpital’s rule 38
Laplace equation 40, 48
polar form 48
Laurent expansion 52
Laurent series 51
Laurent series representation 51
74
left side 25
limit 7, 9
limit superior 12
linear fractional transformation 23
Liouville’s theorem 40
little o notation 64
little Picard theorem 41
local mapping theorem 58
logarithm function 19
logarithm theorem 34
logarithm function 19
principal branch 19
loop theorem 34
maximum modulus theorem 42
maximum/minimum principle 47
mean value theorem 41, 47
for harmonic functions 47
meromorphic 56
minimum modulus theorem 42
M -L inequality 30
Mo¨bius transformation 23
Morera’s theorem 49
multiplicity 41
neighborhood 4
open disk 4
open mapping theorem 42
open set 4
order 41, 52, 54
orientation 25
orientation principle 25
Picard theorem 41, 53
great 53
little 41
piecewise smooth 29
Poisson integral formula 47
polar representation 1
pole 52, 54
order 52, 54
polygon property 33
polygon theorem 34
polygonally connected 4
power series 12
power series representation 37
principal branch 19
principal part 53
principal value 20, 62
of improper integral 62
radius of convergence 13
rational function 7
rectangle 32
rectangle property 32
rectangle theorem 32
region 4
regular 11
removable singularity 52, 54
residue 56, 61
residue theorem 56
Riemann mapping 26
Riemann mapping theorem 26
Riemann sense 62
Riemann sphere 10
Riemann’s principle 53
right side 25
right-angled polygon 5
root 41
root test 12
Rouche´’s theorem 58
schlicht 11
Schwarz reflection principle 50
Schwarz’s lemma 42
simple closed curve 29
simple root 41
simply connected 5
stereographic projection 10
Study’s theorem 45
symmetric points 24
symmetry principle 25
Taylor’s theorem 14
term test 2, 70
for series 2
for infinite product 70
trigonometric functions 19
triple root 41
uniform convergence 7,8
uniqueness theorem 14, 41
for power series 14
univalent 11
upper limit 12
Wallis’ formula 69
Weierstrass factorization theorem 71
Weierstrass M -test 7
Weierstrass’s theorem 49
winding number 36
75

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