MA2325 COMPLEX ANALYSIS
A BRIEF SUMMARY
Contents
1.
2.
3.
4.
5.
Complex numbers
Analytic functions
Power series
Complex integration
Residue theorem and applications
2
5
7
9
12
1
2
MA2325 COMPLEX ANALYSIS A BRIEF SUMMARY
1. Complex numbers
In this section we establish the algebraic properties of complex numbers, geometric
and topological properties in the complex plane and introduce complex function
theory. The set of real numbers is denoted R and
R2 = R × R = {(a, b) : a, b ∈ R}
denotes the set of all ordered pairs of real numbers. The field C of complex numbers
consists of R2 together with two operations:
• addition: (a, b) + (c, d) = (a + c, b + d)
• multiplication: (a, b) · (c, d) = (ac − bd, ad + bc)
The complex number (0, 1) is called the imaginary unit and is denoted i. We identify
each real number a ∈ R with the complex number (a, 0). With these conventions
every complex number (a, b) can be expressed as a + ib.
The mappings
Re : C → R,
z = x + iy 7→ x
Im : C → R,
z = x + iy 7→ y
are called respectively the real part and the imaginary part.
The mapping C → C, z = x + iy 7→ z̄ = x − iy is called the complex conjugate.
Geometrically z̄ is the reflection of z in the realpaxis.
The mapping C → R, z = x + iy 7→ |z| = x2 + y 2 is called the modulus and
represents the distance in the complex plane from the point z to the origin.
Theorem 1.1. The following identities hold for any z, w ∈ C.
(1) The triangle inequality:
|z + w| ≤ |z| + |w|
(2) Reverse triangle inequality:
||z| − |w|| ≤ |z − w|
(3) The parallelogram law:
|z + w|2 + |z − w|2 = 2(|z|2 + |w|2 )
Let z0 ∈ C and r > 0 a positive real number. Then
D(z0 , r) := {z ∈ C : |z − z0 | < r}
is called the open disk with centre z0 and radius r.
Let A ⊂ C and z0 ∈ A. If there exists an open disk D(z0 , r) with centre z0 which
is contained in A then z0 is called an interior point of A. If every point in A is an
interior point of A then A is called an open set.
Let A ⊂ C and z0 ∈ C. If every open disk D(z0 , r) centred at z0 contains a point
which is in A and a point which is not in A then z0 is called a boundary point of A.
If every boundary point of A is contained in A then A is called a closed set.
Note that a subset A ⊂ C is an open set if and only if its complement C\A is a
closed set.
A subset A ⊂ C is called a bounded set if there exists an open disk D(z0 , r) which
contains A.
A subset A ⊂ C is called compact if it is both a closed set and a bounded set.
An open set A ⊂ C is called connected if there does not exist a pair of non-empty
disjoint open sets U , V with A ⊂ U ∪ V .
MA2325 COMPLEX ANALYSIS
A BRIEF SUMMARY
3
A sequence (zn )∞
n=1 of complex numbers is said to be convergent if there exists a
complex number w ∈ C with the property that given any positive real number > 0
there exists a natural number N ∈ N such that |zn − w| < for all n > N . Note
that if such a complex number w exists then it is unique. We call w the limit of the
sequence and write w = limn→∞ zn .
Theorem 1.2. Let (zn )∞
n=1 be a sequence of complex numbers. The following conditions are equivalent.
(1) (zn )∞
n=1 is a convergent sequence.
(2) (zn )∞
n=1 satisfies the Cauchy criterion:
Given any positive real number > 0 there exists a natural number N ∈ N
such that |zm − zn | < for all m, n > N .
∞
(3) (Re(zn ))∞
n=1 and (Im(zn ))n=1 are convergent sequences in R.
Let A ⊂ C and w ∈ C. Then w is called a limit point of A if there exists a
sequence (zn )∞
n=1 of complex numbers in A\{w} with w = limn→∞ zn . Note that a
subset A is a closed set if and only if every limit point of A is contained in A.
Next we introduce limits and continuity for functions which take a complex variable and which are complex-valued.
Definition 1.3. Let A ⊂ C and let f : A → C be a complex-valued function.
Suppose w ∈ C is a limit point of A. A complex number L is called a limit of f as z
approaches w if given any positive real number there exists a positive real number
δ such that
z ∈ A, 0 < |z − w| < δ =⇒ |f (z) − L| < If a limit L exists then it is unique and we write L = limz→w f (z).
Theorem 1.4. If L = limz→w f (z) and M = limz→w g(z) then
(1) limz→w (f + g)(z) = L + M
(2) limz→w (f.g)(z) = LM
(z)
L
(3) limz→w fg(z)
=M
provided M 6= 0.
Let f : A → C be a complex-valued function. The real-valued functions
Re(f ) : A → R,
z 7→ Re(f (z))
Im(f ) : A → R,
z 7→ Im(f (z))
are called respectively the real part of f and the imaginary part of f .
Theorem 1.5. Let A ⊂ C and let f : A → C be a complex-valued function. Suppose
w ∈ C is a limit point of A. Then limz→w f (z) exists if and only if limz→w Re(f )(z)
and limz→w Im(f )(z) both exist. In this case
lim f (z) = lim Re(f )(z) + lim Im(f )(z)
z→w
z→w
z→w
Definition 1.6. Let A ⊂ C and let w ∈ A. A function f : A → C is continuous
at w if limz→w f (z) = f (w). If f is continuous at every point in A then we say f is
continuous on A.
Theorem 1.7. A function f : A → C is continuous at a point w ∈ A if and only if
Re(f ) and Im(f ) are both continuous at w.
Let A ⊂ C and let f : A → C. The preimage of a subset U ⊂ C under f is
f −1 (U ) = {z ∈ A : f (z) ∈ U }
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MA2325 COMPLEX ANALYSIS A BRIEF SUMMARY
Theorem 1.8. Let A ⊂ C be an open set and let f : A → C. The following
statements are equivalent.
(1) f is continuous on A.
(2) The preimage f −1 (U ) of every open set U ⊂ C is an open set in A.
(3) If (zn ) is a sequence in A which converges to w ∈ A then limn→∞ f (zn ) =
f (w).
Theorem 1.9. The composition g◦f of two continuous maps f and g is a continuous
map.
Example 1.10.
(1) Polynomial functions.
p(z) = a0 + a1 z + a2 z 2 + · · · + an z n
These functions are defined and continuous at every point in C.
(2) Rational functions.
a0 + a1 z + a2 z 2 + · · · + an z n
b0 + b1 z + b2 z 2 + · · · + bm z m
These functions are defined and continuous at w ∈ C provided the denominator does not vanish at w.
(3) The exponential function.
f (z) =
Exp : C → C,
z = x + iy 7→ ez = ex (cos y + i sin y)
This function is continuous on C.
(4) Trigonometric functions.
eiz + e−iz
2
iz
e − e−iz
sin z =
2i
These functions are continuous on C.
cos z =
The following are examples of multi-valued functions.
Example 1.11.
(1) nth root
√
n
z = {w : wn = z}
(2) Argument function
arg z = {θ : z = |z|eiθ }
The value of arg z which lies in the range (−π, π] is called the principle value
of the argument and is denoted Arg(z).
(3) Logarithm function
log z = {log |z| + i(Arg(z) + 2nπ) : n ∈ Z}
Definition 1.12. Let A ⊆ C and suppose that for each n ∈ N we have a function
fn : A → C. The sequence of functions (fn )∞
n=1 is said to converge uniformly on A
to a function f : A → C if given any > 0 there exists N ∈ N such that
sup |fn (z) − f (z)| < ,
∀n > N
z∈A
Theorem 1.13 (Uniform Limit Theorem). Let (fn ) be a sequence of continuous
functions converging uniformly on A to f : A → C. Then f is continuous.
MA2325 COMPLEX ANALYSIS
A BRIEF SUMMARY
5
2. Analytic functions
Let A ⊂ C and let w be a limit point of A. Then f : A → C is called complex
differentiable at w if
f (z) − f (w)
lim
z→w
z−w
exists. We call this limit the complex derivative of f (z) at w and denote it by f 0 (w)
df
or dz
(w).
Note that if f (z) is complex differentiable at w then f (z) is continuous at w.
Let A ⊂ C be an open set. A function f : A → C is called analytic on A if it is
complex differentiable at every point in A.
Theorem 2.1. Let f, g : A → C be complex differentiable at w.
(1) Linearity:
(λf + µg)0 (w) = λf 0 (w) + µg 0 (w), ∀ λ, µ ∈ C
(2) Product rule:
(f g)0 (w) = f 0 (w)g(w) + f (w)g 0 (w)
(3) Quotient rule:
0
f
f 0 (w)g(w) − f (w)g 0 (w)
(w) =
g
g(w)2
Theorem 2.2. Let A be an open set, f : A → C and z0 = x0 + iy0 ∈ A. Then
f = u + iv is complex differentiable at z0 if and only if
(1) u and v are both real differentiable at (x0 , y0 ), and,
(2) the Cauchy-Riemann equations are satisfied at (x0 , y0 ):
∂u
∂v
(x0 , y0 ) =
(x0 , y0 )
∂x
∂y
∂u
∂v
(x0 , y0 ) = − (x0 , y0 )
∂y
∂x
Theorem 2.3 (The Chain Rule). Let f : A → C be complex differentiable at z0 and
g : B → C complex differentiable at f (z0 ). Then g ◦ f is complex differentiable at z0
and
(g ◦ f )0 (z0 ) = g 0 (f (z0 ))f 0 (z0 )
Theorem 2.4 (Inverse Function Theorem). Let A be an open set and f : A → C
one-to-one. Suppose f is complex differentiable at z0 , f 0 (z0 ) 6= 0 and the inverse
function g is continuous. Then g is complex differentiable at w0 = f (z0 ) and
g 0 (w0 ) =
1
f 0 (z0 )
Example 2.5.
(1) The square root. We can construct an analytic branch on
the slit plane such that
√
1
( w)0 = √
2 w
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MA2325 COMPLEX ANALYSIS A BRIEF SUMMARY
(2) The Logarithm. We can construct an analytic branch on the slit plane such
that
1
Log 0 (z) =
z
MA2325 COMPLEX ANALYSIS
A BRIEF SUMMARY
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3. Power series
Let (ak )∞
k=0 be a sequence of complex numbers.
P
∞
Definition 3.1. The series ∞
k=0 ak is called convergent if the sequence (sn )n=1 of
partial sums
s n = a0 + a1 + a2 + · · · + an
is convergent in C.
If a series is not convergent it is called divergent.
P
Definition
The series ∞
k=0 ak is called absolutely convergent if the correspondP3.2.
ing series ∞
|a
|
is
convergent
in R.
k
k=0
P∞
Theorem 3.3. If a series k=0 ak is absolutely convergent then it is convergent.
Let A ⊂ C and let fk : A → C be complex-valued functions, k ∈ N.
P
Definition 3.4. The series ∞
k=1 fk (z) is said to be uniformly convergent on A if
∞
the sequence (sn (z))n=1 of partial sums
sn (z) = f1 (z) + f2 (z) + · · · + fn (z)
converges uniformly on A.
In particular, a power series
(sn (z0 ))∞
n=1 of partial sums
P∞
k=0
ak z k is convergent at a point z0 if the sequence
sn (z0 ) = a0 + a1 z0 + a2 z02 + · · · + an z0n
is convergent in C.P(Here (sn (z0 ))∞
n=1 is a sequence of complex numbers).
∞
k
A power series k=0 ak z is absolutely convergent at a point z0 if the sequence
(tn (z0 ))∞
n=1 of partial sums
tn (z0 ) = |a0 | + |a1 z0 | + |a2 z02 | + · · · + |an z0n |
is convergent in R.P
(Here (tn (z0 ))∞
n=1 is a sequence of real numbers).
∞
k
a
z
is
uniformly
convergent on a set A if the sequence
A power series
k=0 k
∞
(sn (z))n=1 of partial sums
sn (z) = a0 + a1 z + a2 z 2 + · · · + an z n
converges uniformly on A. (Here (sn (z))∞
n=1 is a sequence of complex-valued functions on A).
Theorem 3.5 (Weierstrass M-test). Let fk : A → C be complex-valued functions,
k ∈ N. If there exists a sequence (Mk )∞
k=1 of real numbers such that
P∞
(1) k=1 Mk is convergent, and,
(2) supz∈A |fk (z)| ≤ Mk for all k
P
then ∞
k=1 fk (z) is uniformly convergent on A.
P∞
k
Theorem 3.6 (Cauchy-Hadamard theorem). Let
be a power series.
k=0 ak z
Then one of the following conditions holds.
(1) The power series is absolutely convergent at every point in C and uniformly
convergent on every closed disk D(0, r) = {z : |z| ≤ r}.
(2) There exists a real number R > 0 such that the power series is absolutely
convergent in D(0, R), uniformly convergent on every closed disk D(0, r)
contained in D(0, R) and divergent at every point outside D(0, R).
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MA2325 COMPLEX ANALYSIS A BRIEF SUMMARY
(3) The power series is divergent at every point z 6= 0.
R is called the radius of convergence for the power series.
Theorem 3.7 (Formula for the radius of convergence). Let
series. Then
(1)
1
p
R=
lim supn→∞ n |an |
(2)
an R = lim n→∞ an+1 P∞
k=0
ak z k be a power
provided this limit exists.
The above can be easily generalised to power series centred at a point z0 .
Theorem
3.8 (Functions defined by a power series). Let A ⊆ C be an open set and
P∞
suppose k=0 ak (z − z0 )k is a power series which is convergent for all z ∈ A.
(1) The function
∞
X
ak (z − z0 )k
f : A → C, z 7→
k=0
is analytic.
(2) The complex derivative of f (z) is
∞
X
0
kak (z − z0 )k−1
f (z) =
k=1
for all z ∈ A.
(3) The coefficients ak are defined uniquely by
ak =
for all k.
f (k) (z0 )
k!
MA2325 COMPLEX ANALYSIS
A BRIEF SUMMARY
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4. Complex integration
Let f : [a, b] → C be a continuous complex-valued function. We define the integral
Z b
Z b
Z b
Im(f (t)) dt
Re(f (t)) dt + i
f (t) dt :=
a
a
a
Theorem 4.1. Let f : [a, b] → C be a continuous function. Then
Z b
Z b
≤
f
(t)
dt
|f (t)| dt
a
a
A path in C is a continuous function γ : [a, b] → C. If γ(a) = γ(b) then γ is called
a closed path. If γ 0 (t) exists and γ 0 : [a, b] → C is continuous then γ is called a C 1
path. The length of a C 1 path γ : [a, b] → C is
Z b
Length(γ) :=
|γ 0 (t)| dt
a
For convenience we also denote the image γ([a, b]) by γ. If f (z) is continuous on
γ then we define the integral of f (z) over γ to be
Z
Z b
f (z) dz :=
f (γ(t))γ 0 (t) dt
γ
a
Theorem 4.2. Let f : A → C be a continuous function on an open set A.
(1) If γ1 and γ2 are equivalent C 1 paths in A then
Z
Z
f (z) dz =
f (z) dz
γ1
γ2
(2) If γ is a C 1 path in A then
Z
Z
f (z) dz = −
f (z) dz
−γ
γ
(3) If F is a primitive for f and γ : [a, b] → C is a C 1 path in A then
Z
f (z) dz = F (γ(b)) − F (γ(a))
γ
1
(4) If γ is a C path in A then
Z
f (z) dz ≤ Length(γ) sup |f (z)|
z∈γ
γ
(5) Let fn : A → C be continuous functions on A, n ∈ N and let γ be a C 1 path
in A. If the sequence (fn ) converges uniformly on γ to f then
Z
Z
lim
fn (z) dz =
f (z) dz
n→∞
γ
γ
Definition 4.3. Let γ be a C 1 closed path. The index function for γ is defined as
Z
1
1
Indγ : C\γ → C, z0 7→
dz
2πi γ z − z0
Theorem 4.4. Let γ be a C 1 closed path.
(1) Indγ is integer-valued.
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MA2325 COMPLEX ANALYSIS A BRIEF SUMMARY
(2) Indγ is continuous on C\γ.
(3) Indγ is constant on each connected component of C\γ.
Note that the integer Indγ (z) is called the winding number of γ about the point z
and measures the number of times the path γ winds around z.
Lemma 4.5. Let F1 ⊃ F2 ⊃ F3 ⊃ · · · be T
a decreasing sequence of non-empty closed
sets in C. If limn→∞ diam(Fn ) = 0 then ∞
n=1 Fn 6= ∅.
Theorem 4.6 (Cauchy’s Theorem Version 1). Let f : A → C be analytic on an
open set A. If T is a triangular path with interior contained in A then
Z
f (z) dz = 0
T
Theorem 4.7 (Cauchy’s Theorem Version 2). Let f : A → C be analytic on a
star-shaped open set A. If γ is a piecewise C 1 closed path in A then
Z
f (z) dz = 0
γ
Theorem 4.8. Let f : A → C be analytic on an open set A and suppose A contains
the closed disk D(w0 , R). If z0 ∈ D(w0 , R) then
Z
Z
f (ζ)
f (ζ)
dζ =
dζ
|ζ−z0 |=r ζ − z0
|ζ−w0 |=R ζ − z0
for all r < R − |z0 − w0 |.
Theorem 4.9 (Cauchy’s integral formula). Let f : A → C be analytic on an open
set A and suppose A contains the closed disk D(w0 , R). If z0 ∈ D(w0 , R) then
Z
1
f (ζ)
f (z0 ) =
dζ
2πi |ζ−w0 |=R ζ − z0
Theorem 4.10 (Local power series representation for analytic functions). Let f :
A → C be analytic on an open set A. If D(z0 , R) is contained in A then
∞
X
f (z) =
ak (z − z0 )k
k=0
for all z ∈ D(z0 , R) where
1
ak =
2πi
Z
|ζ−z0 |=r
f (ζ)
dζ
(ζ − z0 )k+1
and 0 < r < R.
Theorem 4.11. Let f : A → C be analytic on an open set A. If D(z0 , R) is
contained in A then
(1)
Z
k!
f (ζ)
(k)
f (z0 ) =
dζ
2πi |ζ−z0 |=r (ζ − z0 )k+1
where 0 < r < R
(2) Cauchy’s estimate:
k!
|f (k) (z0 )| ≤ k sup |f (z)|
R z∈D(z0 ,R)
MA2325 COMPLEX ANALYSIS
A BRIEF SUMMARY
11
A function f : A → C is called bounded if there exists a real number M such that
|f (z)| ≤ M for all z ∈ A.
A function which is analytic on the whole of the complex plane C is called entire.
Theorem 4.12 (Liouville’s Theorem). Every bounded entire function is constant.
Theorem 4.13 (Fundamental Theorem of Algebra). For every non-constant polynomial p(z) there exists z0 ∈ C with p(z0 ) = 0.
Theorem 4.14 (The Identity Theorem). Let f : A → C and g : A → C be analytic
on an open and connected set A. If f (z) = g(z) on a set which has a limit point in
A then f (z) = g(z) for all z ∈ A.
Theorem 4.15 (Uniform limit of analytic functions). Let A be open and fk : A → C
analytic for each k ∈ N. If the sequence (fk (z)) converges locally uniformly on A to
a function f : A → C then f is analytic.
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MA2325 COMPLEX ANALYSIS A BRIEF SUMMARY
5. Residue theorem and applications
sequence of complex numbers and let z0 ∈ C.
Let (ak )∞
k=−∞
P∞ be a doubly infinite
k
The series
a Laurent series centred at z0 . We call
k=−∞ ak (z − z0 ) is called
P∞
P∞
k
−k
the principal part of the
k=0 ak (z − z0 ) the analytic part and
k=1 a−k (z − z0 )
Laurent series.
P
k
Theorem 5.1 (Convergence properties for a Laurent series). Let ∞
k=−∞ ak (z−z0 )
be a Laurent series. Then there are two possibilities:
(1) The Laurent series is divergent at every point in C.
(2) There exists an annulus A = {z : R1 < |z − z0 | < R2 } such that the Laurent
series is absolutely convergent at every point in A, uniformly convergent on
every closed annulus {z : r1 ≤ |z − z0 | ≤ r2 } where R1 < r1 < r2 < R2 and
divergent at every point outside the closed annulus {z : R1 ≤ |z − z0 | ≤ R2 }
Theorem
5.2 (Functions defined by a Laurent series). Let A be an open set and
P∞
suppose k=−∞ ak (z − z0 )k is a Laurent series which is convergent for all z ∈ A.
(1) The function
f : A → C,
z 7→
∞
X
ak (z − z0 )k
k=−∞
is analytic.
(2) The coefficients ak are defined uniquely by
Z
f (ζ)
1
ak =
dζ
2πi |ζ−z0 |=r (ζ − z0 )k+1
for all k ∈ Z where R1 < r < R2 .
If a function f (z) is analytic on a punctured disk D(z0 , r)\{z0 } then z0 is called
an isolated singularity for f (z).
Lemma 5.3. Let f : A → C be analytic on an open set A. Then the function
f (z)−f (z0 )
, z 6= z0
z−z0
g : A → C, z 7→
0
f (z0 ),
z = z0
is analytic.
Theorem 5.4. If f (z) has an isolated singularity at z0 then there exists a punctured
disk D(z0 , R)\{z0 } such that for any 0 < r1 < r2 < R,
Z
Z
f (ζ)
1
f (ζ)
1
dζ −
dζ
f (z) =
2πi |ζ−z0 |=r2 ζ − z
2πi |ζ−z0 |=r1 ζ − z
for all z contained in the annulus {z : r1 < |z − z0 | < r2 }.
Theorem 5.5 (Local Laurent series representation for analytic functions). If f (z)
has an isolated singularity at z0 then f (z) has a Laurent series representation
f (z) =
∞
X
ak (z − z0 )k
k=−∞
for all z in some punctured disk D(z0 , r)\{z0 }.
MA2325 COMPLEX ANALYSIS
A BRIEF SUMMARY
13
Let z0 be an isolated singularity for f (z) and let
∞
X
f (z) =
ak (z − z0 )k
k=−∞
be the Laurent series representation for f (z) centred at z0 . Then z0 is called
(1) a removable singularity if ak = 0 for all k < 0,
(2) a pole if ak = 0 for all k < N some N < 0,
(3) an essential singularity if ak 6= 0 for infinitely many k < 0.
The coefficient a−1 is called the residue of f at z0 , denoted Res(f, z0 ).
Theorem 5.6 (The Residue Theorem). Let A be an open and star-shaped set in C.
Let f : A\{z1 , . . . , zn } → C be analytic with isolated singularities at {z1 , . . . , zn }. If
γ is a piecewise C 1 closed path in A\{z1 , . . . , zn } then
" n
#
Z
X
f (z) dz = 2πi
Res(f, zj ) Indγ (zj )
γ
j=1
Example 5.7. Contour integration.
Z ∞
π
1
√
dx
=
x4 + 1
2
−∞
Theorem 5.8 (Jordan’s Lemma). Let CR : [0, π] → C, t 7→ Reit be a semicircular
path and suppose f (z) is continuous on CR for R large. If
lim
sup |f (z)| = 0
R→∞
then
z∈CR
Z
iλz
lim
f (z)e
R→∞
dz
CR
for all λ > 0.
Example 5.9. Contour integration.
Z ∞
−∞
π
cos x
dx = 2
2
x +4
2e
=0