Complex Analysis
1 Holomorphic functions
1.1 Definitions
Def 1.1.1: Let U, V ⊂ C be open (often, we can take V = C) and let a ∈ U . A function
f : U → V is complex differentiable in a if
f (z) − f (a)
f ′ (a) = lim
z→a z−a
exists. We can reformulate this: f is complex differentiable in a if and only if there exists a
f ′ (a) = w ∈ C such that
f (z) − f (a) − w · (z − a) z→a
−→ 0
|z − a|
Def 1.1.1.a: f : U → V is called holomorphic if it is complex differentiable in all a ∈ U . An
entire function is a holomorphic function that is defined on the whole plane, i.e. f : C → C.
Proposition 1.1.3: Let U ⊂ C be open, let a ∈ U , c ∈ C and let f, g : U → C be complex
differentiable in a. Then the following properties hold:
• (f ± g)′ (a) = f ′ (a) ± g ′ (a)
• (c · f )′ (a) = c · f ′ (a)
• (f g)′ (a) = f ′ (a) · g(a) + f (a) · g ′ (a) (Leibnitz rule)
−f ′ (a)
• if f (a) ̸= 0 and f ′ (a) ̸= 0, then ( f1 )′ (a) = f 2 (a)
(inversion rule)
• Let V ⊂ C be open such that im(f ) ⊂ V . Take a function h : V → C which is complex
differentiable in f (a). Then (h ◦ f )′ (a) = h′ (f (a)) · f ′ (a) (chain rule)
1.2 The Cauchy-Riemann equation
Given a function f : U → C (or we can identify C = R2 ). Then there exists uniquely determined
functions u : U → R and v : U → R where (x, y) 7→ u(x, y) and (x, y) 7→ v(x, y) such that
f (x + iy) = u(x, y) + i · v(x, y) (i.e. u(x, y) = Re(f (x + iy)) and v(x, y) = Im(f (x + iy)))
Def 1.2.1: Take a = (b, c) ∈ U ⊂ C = R2 and let f : U → R2 . Then f is (real) differentiable
in a if there exists a 2 × 2 matrix Jf (a) (the Jacobian matrix of f ) such that
f (x, y) − f (b, c) − Jf (a) · ((x, y) − (b, c))
lim = (0, 0)
(x,y)→(b,c) ||(x, y) − (b − c)||
Whats the difference between real differentiable and complex differentiable? Complex is much
stronger. Complex differentiability is a certain restriction on the Jacobian matrix
1
,Lemma 1.2.2: A ∈ R2×2 ,w = r+ is ∈ C. Then we have that w · z = A · (x, y) for all
r −s
z = x + iy if and only if A =
s r
Let f : U → C for U ⊂ C open and f (x + iy) = u(x, y) + i · v(x, y). In the future, we simply
write f = u + iv.
Theorem 1.2.3Let f : u + iv : U → C. Then f is holomorphic if and only if f is differentiable
and if the Cauchy-Riemann equations are satisfied: ux = vy and uy = −vx .
If f is holomorphic, then f ′ (a) = ux (a) + ivx (a) for all a ∈ U .
Reminder: f is differentiable in a = (b, c) if there exists Jf (a) ∈ R2×2 such that
f (x, y) − f (b, c) − Jf (a)[(x, y) − (b, c)]
lim =0
(x,y)→(b,c) ||(x, y) − (b, c)||
ux uy
we write Jf (a) =
vx vy
Theorem 1.2.5 (inverse function theorem) Let U, V ⊂ C be open and let f : U → V be a
holomorphic bijection with inverse bijection g : V → U , If g is continuous and if f ′ (a) ̸= 0 for
all a ∈ U , then g is holomorphic with derivative for all b ∈ V :
1
f ′ (b) =
f ′ (g(b))
Note: the assumption that g is continuous is not necessary.
1.3 Power series
Definition 1.3.1 A (complex) power series is an expression of the form
∞
X
ai z i
i=0
ai z i for short.
P
with a0 , a1 , · · · ∈ C. When the context is clear, we write
ai z i = f (z) means that ai z i converges (i.e. the sequence of partial sums
P P
When we write
converges) to f (z) for ”some given” z ∈ C
P i
z converges absolutely ( |z i | converges. Absolute
P
Lemma 1.3.2 The geometric series
convergence implies convergence) for all z ∈ D1 (0):
X 1
zi =
1−z
ai z i is given by
P
Definition 1.3.3 The radius of convergence of a power series
1 1 1
R= , = ∞, =0
lim sup |an |1/n 0 ∞
We have that
lim sup bn = lim (sup{bn | n ≥ N })
N →∞
2
, TheoremP1.3.4 (Cauchy-Hadamard) Let ai z i be a power series with radius of convergence
P
R. Then ai z i converges absolutely if |z| < R and it diverges if |z| > R. For |z| = R it depends
on the series and specific z whether the series converges or diverges.
Exponential series:
∞
X 1 n
ez = z
n!
n=0
Cosine series: X in n
cos(z) = z
n!
n∈N even
Sin series:
X in−1 n
sin(z) = z
n!
n∈N odd
an z n be a power series with radius of convergence R. Then the function
P
Theorem 1.3.7 Let
f : DR (0) → C
X
z 7→ an z n
Is holomorphic with derivative
∞
X
f ′ (z) = (n + 1) · an+1 z n
n=0
which is a power series with the same radius of convergence R.
Corollary 1.3.8 The functions exp , cos and sin are entire.
an z n and g(z) = bn z n that are
P P
Proposition 1.3.9 Consider two power series f (z) =
absolutely convergent on Dr (0) with r > 0 and let c ∈ C. Then
∞
X
(f + g)(z) = (an + bn )z n
n=0
∞
X
(c · f )(z) = (c · an )z n
n=0
∞
X X
(f · g)(z) = ak bl z n (Cauchy product)
n=0 k,l≥0 s.t.k+l=n
for all z ∈ Dr (0). In particular, all power series converge absolutely for z ∈ Dr (0)
Proposition 10 For all z = x + iy, w ∈ C:
• eiz = cos(z) + i sin(z) (Eu-• ez = ex (cos(y) + i sin(y)) • cos′ (z) = − sin(z)
ler’s formula)
• sin′ (z) = cos(z)
• ez+w = ez · ew
• cos(z) = 21 (eiz + e−iz )
• z = |z|·ei arg z where arg z ∈
• sin(z) = 1
2π (e
iz − e−iz ) • exp′ (z) = exp(z) R/2π · Z
3
1 Holomorphic functions
1.1 Definitions
Def 1.1.1: Let U, V ⊂ C be open (often, we can take V = C) and let a ∈ U . A function
f : U → V is complex differentiable in a if
f (z) − f (a)
f ′ (a) = lim
z→a z−a
exists. We can reformulate this: f is complex differentiable in a if and only if there exists a
f ′ (a) = w ∈ C such that
f (z) − f (a) − w · (z − a) z→a
−→ 0
|z − a|
Def 1.1.1.a: f : U → V is called holomorphic if it is complex differentiable in all a ∈ U . An
entire function is a holomorphic function that is defined on the whole plane, i.e. f : C → C.
Proposition 1.1.3: Let U ⊂ C be open, let a ∈ U , c ∈ C and let f, g : U → C be complex
differentiable in a. Then the following properties hold:
• (f ± g)′ (a) = f ′ (a) ± g ′ (a)
• (c · f )′ (a) = c · f ′ (a)
• (f g)′ (a) = f ′ (a) · g(a) + f (a) · g ′ (a) (Leibnitz rule)
−f ′ (a)
• if f (a) ̸= 0 and f ′ (a) ̸= 0, then ( f1 )′ (a) = f 2 (a)
(inversion rule)
• Let V ⊂ C be open such that im(f ) ⊂ V . Take a function h : V → C which is complex
differentiable in f (a). Then (h ◦ f )′ (a) = h′ (f (a)) · f ′ (a) (chain rule)
1.2 The Cauchy-Riemann equation
Given a function f : U → C (or we can identify C = R2 ). Then there exists uniquely determined
functions u : U → R and v : U → R where (x, y) 7→ u(x, y) and (x, y) 7→ v(x, y) such that
f (x + iy) = u(x, y) + i · v(x, y) (i.e. u(x, y) = Re(f (x + iy)) and v(x, y) = Im(f (x + iy)))
Def 1.2.1: Take a = (b, c) ∈ U ⊂ C = R2 and let f : U → R2 . Then f is (real) differentiable
in a if there exists a 2 × 2 matrix Jf (a) (the Jacobian matrix of f ) such that
f (x, y) − f (b, c) − Jf (a) · ((x, y) − (b, c))
lim = (0, 0)
(x,y)→(b,c) ||(x, y) − (b − c)||
Whats the difference between real differentiable and complex differentiable? Complex is much
stronger. Complex differentiability is a certain restriction on the Jacobian matrix
1
,Lemma 1.2.2: A ∈ R2×2 ,w = r+ is ∈ C. Then we have that w · z = A · (x, y) for all
r −s
z = x + iy if and only if A =
s r
Let f : U → C for U ⊂ C open and f (x + iy) = u(x, y) + i · v(x, y). In the future, we simply
write f = u + iv.
Theorem 1.2.3Let f : u + iv : U → C. Then f is holomorphic if and only if f is differentiable
and if the Cauchy-Riemann equations are satisfied: ux = vy and uy = −vx .
If f is holomorphic, then f ′ (a) = ux (a) + ivx (a) for all a ∈ U .
Reminder: f is differentiable in a = (b, c) if there exists Jf (a) ∈ R2×2 such that
f (x, y) − f (b, c) − Jf (a)[(x, y) − (b, c)]
lim =0
(x,y)→(b,c) ||(x, y) − (b, c)||
ux uy
we write Jf (a) =
vx vy
Theorem 1.2.5 (inverse function theorem) Let U, V ⊂ C be open and let f : U → V be a
holomorphic bijection with inverse bijection g : V → U , If g is continuous and if f ′ (a) ̸= 0 for
all a ∈ U , then g is holomorphic with derivative for all b ∈ V :
1
f ′ (b) =
f ′ (g(b))
Note: the assumption that g is continuous is not necessary.
1.3 Power series
Definition 1.3.1 A (complex) power series is an expression of the form
∞
X
ai z i
i=0
ai z i for short.
P
with a0 , a1 , · · · ∈ C. When the context is clear, we write
ai z i = f (z) means that ai z i converges (i.e. the sequence of partial sums
P P
When we write
converges) to f (z) for ”some given” z ∈ C
P i
z converges absolutely ( |z i | converges. Absolute
P
Lemma 1.3.2 The geometric series
convergence implies convergence) for all z ∈ D1 (0):
X 1
zi =
1−z
ai z i is given by
P
Definition 1.3.3 The radius of convergence of a power series
1 1 1
R= , = ∞, =0
lim sup |an |1/n 0 ∞
We have that
lim sup bn = lim (sup{bn | n ≥ N })
N →∞
2
, TheoremP1.3.4 (Cauchy-Hadamard) Let ai z i be a power series with radius of convergence
P
R. Then ai z i converges absolutely if |z| < R and it diverges if |z| > R. For |z| = R it depends
on the series and specific z whether the series converges or diverges.
Exponential series:
∞
X 1 n
ez = z
n!
n=0
Cosine series: X in n
cos(z) = z
n!
n∈N even
Sin series:
X in−1 n
sin(z) = z
n!
n∈N odd
an z n be a power series with radius of convergence R. Then the function
P
Theorem 1.3.7 Let
f : DR (0) → C
X
z 7→ an z n
Is holomorphic with derivative
∞
X
f ′ (z) = (n + 1) · an+1 z n
n=0
which is a power series with the same radius of convergence R.
Corollary 1.3.8 The functions exp , cos and sin are entire.
an z n and g(z) = bn z n that are
P P
Proposition 1.3.9 Consider two power series f (z) =
absolutely convergent on Dr (0) with r > 0 and let c ∈ C. Then
∞
X
(f + g)(z) = (an + bn )z n
n=0
∞
X
(c · f )(z) = (c · an )z n
n=0
∞
X X
(f · g)(z) = ak bl z n (Cauchy product)
n=0 k,l≥0 s.t.k+l=n
for all z ∈ Dr (0). In particular, all power series converge absolutely for z ∈ Dr (0)
Proposition 10 For all z = x + iy, w ∈ C:
• eiz = cos(z) + i sin(z) (Eu-• ez = ex (cos(y) + i sin(y)) • cos′ (z) = − sin(z)
ler’s formula)
• sin′ (z) = cos(z)
• ez+w = ez · ew
• cos(z) = 21 (eiz + e−iz )
• z = |z|·ei arg z where arg z ∈
• sin(z) = 1
2π (e
iz − e−iz ) • exp′ (z) = exp(z) R/2π · Z
3
