Complex analysis and its real life application notes

1. Introductory Complex Analysis Cheat Sheet
We construct the field of complex numbers as the following quotient ring,
C = R[x]/hx2
+ 1i
Algebra of Complex Numbers
• Addition: (a + ib) + (c + id) = (a + c) + i(b + d)
• Multiplication: (a + ib)(c + id) = (ac − bd) + i(ad + bc)
• Division:
a + ib
c + id
=
(ac + bd) + i(bc − ad)
c2 + d2
• Square root:
√
a + ib = ±
r
a +
√
a2 + b2
2
+ i
b
|b|
r
−a +
√
a2 + b2
2
!
• <(a + ib) = a, =(a + ib) = b
Field of Complex Numbers
• Complex conjugation: a + ib = a − ib
– a + b = a + b
– ab = a · b
Geometrically, conjugation is reflection over the real axis.
• Absolute value: |a| = +
√
aa
– |ab| = |a| · |b|
– |a + b|2
= |a|2
+ |b|2
+ 2<(ab)
– |a − b|2
= |a|2
+ |b|2
− 2<(ab)
– |a + b|2
+ |a − b|2
= 2(|a|2
+ |b|2
)
The absolute value function forms the metric on C. C is complete under
this metric.
Conjugation, Absolute Value
Some basic results:
• For z0 ∈ C, r > 0 we denote the ball (i.e. disk) of radius r around z0 to
be B(z0, r) = {z ∈ C | |z − z0| < r}
• A point z ∈ C is a limit point of E ⊆ C if ∀ε > 0, B(z, ε) ∩ E contains
a point other than z.
• A subset E ⊆ C is said to be open if ∀z ∈ E, ∃ r > 0, s.t. B(z, r) ⊂ E.
• A subset E ⊆ C is said to be closed, if CE is open in C. Or equivalently
a set which contains all its limit points.
Some properties of open sets:
• C and Ø are open subsets of C.
• All finite intersections of open sets are open sets.
• The collection of all open sets on C form a topology on C.
Interior, closure, density
• Interior: Let E ⊆ C. The interior of E is defined as, E◦
=set of all
interior points of E, or equivalently, ∪{Ω | Ω ⊆ E ∧ Ω is open in C}
• Closure: Let E ⊆ C. The closure of E is defined as ˆ
{F | E ⊆
F ∧ F is closed in C}
• Density: Let E ⊆ D, the closure of E in D is D. Then E is called dense
in D.
Path : A path in a metric space from a point x ∈ X to y ∈ Y is a continuous
mapping γ : [0, 1] → X s.t. γ(0) = x and γ(1) = y.
Separated and Connected
For a metric space (X, d).
• Separated: X is separated if ∃ disjoint non-empty open subsets A, B
of X s.t. X = A ∪ B.
• Connected:
– X is connected if it has no separation.
– X is connected ⇐⇒ X does not contain a proper subset of X
which is both open and closed in X.
– Continuous functions preserve connectedness.
– An open subset Ω ∈ C is connected ⇐⇒ for z, w ∈ Ω, there exists
a path from z to w.
Basic Topological definitions in C
Open cover: Let (X, d) be a metric space and E be a collection of open sets in
X. We say that U is an open cover of a subset K ⊆ X, if K ⊂
S
{U | U ∈ E}
Compactness: For some K ⊆ X is compact if for every open cover E of
K, there exists E1, · · · .En ∈ E s.t. K ⊂ Un
i=1En, i.e. it is compact if it has a
finite open cover.
• In a metric space, a compact set is closed.
• A closed subset of a compact set is closed.
Limit point compact: We say a metric space X is limit point compact if
every infinite subset of X has a limit point.
• If X is a compact metric space, then it is also limit point compact.
Sequentially compact: We say a metric space X is sequentially compact if
every sequence has a convergent sub-sequence.
• If X is limit point compact then X is sequentially compact.
• Let X be sequentially compact, then X is a compact metric space.
Lebesgue number lemma: Let X be sequentially compact, and let U be
an open cover of X. Then ∃ δ > 0 s.t. for x ∈ X, ∃ u ∈ U s.t. B(x, δ) ⊆ u.
Basic Topological definitions in C contd.
A function f : C → C is called an isometry if |f(z)−f(w)| = |z −w| , ∀z, w ∈
C.
• Let f be an isometry s.t. f(0) = 0, then the inner product hf(z), f(w)i =
hz, wi , ∀z, w ∈ C.
• If f is an isometry s.t. f(0) = 0 then f is a linear map.
• The standard argument for a + ib ∈ C, Arg(a + ib) = tan−1 b
a
Isometries on the Complex Plane
Uniform convergence: Let Ω ⊆ C and f1, · · · , fn : Ω → C be a set of func-
tions on Ω. We say, {f}n∈N converges uniformly to f if given ε > 0, ∃n ∈ N
s.t. |fn(x) − f(x)| < ε, ∀x ∈ Ω and n ≥ N.
Complex exponential: For z ∈ C, exp(z) =
P∞
n=0
zn
n!
Trigonometric functions: For z ∈ C, cos(x) = eiz
+e−iz
2 and sin(x) = eiz
−eiz
2
Hyperbolic trigonometric functions: For z ∈ C, cosh(x) = ez
+e−z
2 and
sinh(z) = ez
−e−z
2
Functions on the Complex Plane
Complex derivative: Let Ω ⊆ C and f : Ω → C, we say that f is complex
differentiable at a point z0 ∈ Ω if z0 is an interior point and the following limit
exists limz→z0
f(z)−f(z0)
z−z0
. The limit is denoted as f0
(z0) or df(z)
dz .
Holomorphic functions: If f : Ω → C is complex differentiable at every
point z ∈ Ω, then f is said to be a holomorphic on Ω. Entire function:
Functions which are complex differentiable on C are called entire functions.
• Complex differentiability implies continuity.
• Complex derivatives of a function are linear transformations.
• Product rule: If f, g : Ω → C are complex differentiable at z0 ∈ Ω. Then
fg is complex differentiable at z0 with derivative f0
(z0)g(z0)+g0
(z0)f(z0).
• Quotient rule: If f, g : Ω → C are complex differentiable at z0 ∈ Ω,
and g doesn’t vanish at z0. Then
f
g
0
(z0) = f0
(z0)g(z0)−g0
(z0)f(z0)
g(z0)2
• Chain rule: If f : Ω → C and g : D → C are complex differentiable at
z0 ∈ Ω, and f(Ω) ⊆ D. Then g(f(x))0
(z0) = g0
(f(z0))f0
(z0)
Complex differentiability
Formal Power Series: A formal power series around z0 ∈ C is a formal
expansion
P∞
n=0 an(z − z0)n
, where an ∈ C and z is indeterminate.
Radius of convergence: For a formal power series
P
an(z − z0)n
the radius
of convergence R ∈ [0, ∞] given by R = lim infn→∞ |an|−1/n
. Using the ratio
test is identical i.e. R = lim infn→∞
|an|
|an+1| .
• The series converges absolutely when z ∈ B(z0, R), and for r R, the
series converges uniformly, else if |z − z0| R the series diverges.
• Let z ∈ C s.t. |z − z0| R, then ∃ infinitely many n ∈ N s.t.
|an|−1/n
|z − z0|.
Abel’s Theorem: Let F(z) =
P∞
n=0 an(z − z0)n
be a power series with a
positive radius of convergence R, suppose z1 = z0 + Reiθ
be a point s.t. F(z1)
converges. Then limr→R− F(z0 + reiθ
) = F(z1)
Power Series
Let F(z) =
P∞
n=1 an(z − z0)n
be a power series around z0 with a radius of
convergence R. Then F is holomorphic in B(z0, R).
• F(x)0
=
P∞
n=1 nan(z − z0)n−1
with same radius of convergence R.
• an = F n
(z0)
n!
Cauchy product of two power series: For power series F(z) =
P
an(z −
z0)n
and G(z) =
P
an(z − z0)n
with degree of convergence at least R. Then
the Cauchy product F(z)G(z) =
P
cn(z − z0)n
where cn =
Pn
k=0 akbn−k also
has degree of convergence at least R.
Differentiation of Power Series
For a complex function f(z) = u(z) + iv(z),
f0
(x) = ∂u
∂x + i∂v
∂x or f0
(x) = −i∂u
∂y + ∂v
∂y
Therefore, we get the two Cauchy-Riemann Differential equations,
•
∂u
∂x
=
∂v
∂y
•
∂u
∂y
= −
∂v
∂x
A function is holomorphic iff it satisfies the Cauchy-Riemann equations.
Wirtinger derivatives:
•
∂
∂z
=
1
2
∂
∂x
+
1
i
∂
∂y
•
∂
∂z
=
1
2
∂
∂x
−
1
i
∂
∂y
If f is holomorphic at z0 then, ∂f
∂z = 0 and f0
(z0) = ∂f
∂z (z0) = 2∂u
∂z (z0)
Cauchy-Riemann Differential Equations
Laplacian: Define ∆ = ∂2
∂x2 + ∂2
∂y2 .
Harmonic function: Let u : Ω → R be a twice differentiable function. We
say that u is a harmonic function if ∆u = 0
For any holomorphic function f, (f), =(f) are examples of harmonic
functions, but there are harmonic functions which are not holomorphic.
Boundary of a set: For a metric space X, Ω ∈ X,
the boundary of Ω = ∂Ω = Ω ∩ ΩC
Maximum principle for harmonic functions: Let u : Ω → R be a twice
differentiable harmonic function. Let k ⊂ Ω be a compact sub set of Ω. Then,
supz∈k u(z) = supz∈∂k u(z) and infz∈k u(z) = infz∈∂k u(z)
Maximum principle for holomorphic functions: Let Ω ⊆ C be open and
connected and let f : Ω → C be a holomorphic function. Then, for compact
k ⊆ Ω, we have, supz∈k |f(z)| = sup∂k |f(z)|
Harmonic conjugate: Let u : Ω → R be a twice differentiable harmonic
function. We say that v : Ω → R is a harmonic conjugate of u if f = u + iv is
holomorphic.
• For a harmonic function from C to R there exists a uniquely determined
harmonic conjugate from C to R (up to constants).
Harmonic Functions

2. Extended complex plane: b
C = C
S
{∞}
Consider S2
, associate every point z = x + iy with a line L that connects to
the point P = (0, 0, 1). L = (1 − t)z + tP, where t ∈ R.
The point at which L for some z touches S2
is given as
2x
|z|2 + 1
,
2y
|z|2 + 1
,
|z|2
− 1
|z|2 + 1
, associate P with ∞. This gives a stereo-
graphic projection of the complex plane unto S2
. This sphere is known as the
Riemann sphere.
Riemann Sphere
A map S(z) =
az + b
cz + d
for a, b, c, d ∈ C is called a Möbius transformation if
ad − bc 6= 0.
Every mobius transformation is holomorphic at C {−d/c}, i.e. every point
other than is zero.
• The set of all mobius transformations is a group under transposition.
• S forms a bijection with b
C
Every mobius transformation can be written as composition of,
1. Translation: S(z) = z + b, b ∈ C
2. Dilation: S(z) = az, a 6= 0, a = eiθ
3. Inversion: S(z) = 1/z
Möbius transformations
A continuous parametrized curve is a continuous map γ : [a, b] → C for a, b ∈ R.
• If a = b the curve is trivial.
• γ(a) is initial point and γ(b) is terminal point.
• γ is said to be closed if γ(a) = γ(b).
• γ is said to be simple if it is injective, i.e. doesn’t ”cross” itself.
• A curve −γ is a reversal of γ if γ : [−a, −b] → C and if −γ(t) = γ(−t)
• γ is said to be continuously differentiable if γ0
(t0) (defined usually) exists
and is continuous.
Reparametrization: We say a curve γ2 : [a2, b2] → C is a continuous
reparametrization of γ1 : [a1, b1] → C, if there exists a homeomorphism ϕ :
[a1, b1] → [a2, b2] s.t.ϕ(a1) = a2, ϕ(b1) = b2 and γ2(ϕ(t)) = γ1(t)∀t ∈ [a1, b1].
• Reparametrization is an equivalence relation.
Arc length: Arc length of curve γ = |γ| = sup
Pn
i=0 |γ(xi+1 − γ(xi))| for all
partitions of [a, b].
• A curve that has a finite arc length is called rectifiable.
• |γ| =
ˆ b
a
|γ0
(t)| dt
Curves in C
Let f : Ω → C be a continuous function. Let F : Ω → C be called the anti-
derivative of f, i.e. F is holomorphic in Ω and F0
(z) = f(z), ∀z ∈ Ω. For a
rectifiable curve γ,
´
γ
f(z)dz = F(z1) − F(z0), where z0 is the initial point
and z1 is the terminal point.
First Fundamental Theorem of Calculus
Let f : Ω → C be a continuous function such that
´
γ
f = 0. Whenever γ is a
closed polygonal path contained in Ω. For fixed z0 ∈ Ω, define a path γ1 from
z0 to z1 such that F(z1) =
´
γ1
f(z) dz. Then F is a well defined holomorphic
function s.t. F0
(z1) = f(z1) ∀z1 ∈ Ω
Second Fundamental Theorem of Calculus
For continuously differentiable curves γ : [a, b] → C, and σ : [b, c] → C
• For a reparametrization b
γ of γ we can say that
´
γ
f(z) dz =
´
b
γ
f(z) dz
•
´
−γ
f(z) dz = −
´
γ
f(z) dz
•
´
γ+σ
f(z) dz =
´
γ
f(z) dz +
´
σ
f(z) dz
•
´
γ
f(z) dz =
´ b
a
f(γ(t))γ0
(t) dt
• If f is bounded by M then
´
γ
f(z) dz ≤ M|γ|
• For c ∈ C, we have,
´
γ
(cf + g)(z) dz = c
´
γ
f(z) dz +
´
γ
g(z) dz
Properties of complex integration
Consider two curves γ0, γ1 → Ω with the same initial and end point [a, b].
We say that γ0 is homotopic to γ1 (γ0 ∼ γ1) if there exists a continuous map
H : [0, 1] × [a, b] → Ω s.t. H(0, t) = γ0(t) and H(1, t) = γ1(t), ∀t ∈ [a, b].
H(s, a) = z0, H(s, b) = z1 ∀s ∈ [0, 1]
For closed curves γ0 at z0 and γ1 at z1, we say that γ0 is homotopic to γ1
as closed curves if there exists a continuous map H : [0, 1] × [a, b] → Ω, s.t.
H(0, t) = γ0(t), H(1, t) = γ1(t), ∀t ∈ [a, b]. And H(s, a) = H(s, b), ∀s ∈ [0, 1].
• Homotopy is an equivalence relation.
Homotopy of curves
Cauchy-Goursat theorem: If a curve γ0 is homotopic to a reparametriza-
tion of γ1 then, the integral of some function f : Ω → C is homotopy invariant,
i.e.,
ˆ
γ0
f =
ˆ
γ1
f
Alternative statement: Let f : Ω → C be holomorphic on Ω, and
γ0 : [a, b] → Ω is a rectifiable curve which is null-homotopic (i.e. homotopic to
a constant map). Then,
ˆ
γ0
f(z) dz = 0
Cauchy-Goursat Theorem
Let Ω ⊆ C be a convex and open set and f : Ω → C be holomorphic on Ω.
Then f has an anti derivative F on Ω, and if γ is a closed rectifiable curve on
Ω then
´
γ
f = 0.
Cauchy’s theorem for convex domains
Let f : Ω → C be holomorphic. Fix z0 ∈ Ω and let r 0 be s.t. B(z0, r) ⊆ Ω.
Suppose γ is a closed curve in Ω{z0} s.t. γ is homotopic to a reparametrization
to γ1 where γ1(t) = z0 + reit
for t ∈ [0, 2π]. Then,
f(z0) =
1
2πi
ˆ
γ
f(z)
z − z0
dz
Cauchy’s integral formula
An alternative statement, we say f : Ω → C is complex analytical if given
z0 ∈ Ω, ∃ B(z0, r) ⊆ Ω s.t. the formal power series
P∞
n=0 an(z −z0)n
converges
in B(z0, r) to f.
Let f : Ω → C be holomorphic on Ω. Suppose for z0 ∈ Ω, B(z0, r) ⊂ Ω, then for
every n ∈ N, let an = 1
2πi
´
γ
f(z)
(z−z0)n+1 dz where γ(t) = z0 + reit
for t ∈ [0, 2π].
Then the power series
P∞
n=0 an(z − z0)n
converges in B(z0, r) to f(z).
Corollary: If f : Ω → C is holomorphic then f0
is also holomorphic. Therefore
f is infinitely differentiable.
Complex analytic function
For a analytic function f : Ω → C s.t. f(z0) = 0 at z0 ∈ Ω, ∃ a unique analytic
function g : Ω → C s.t. f(z) = (z − z0)g(z)
Factor theorem for analytic function
• Let Ω be open and connected subset of C. and f, g : Ω → C be analytic
functions on Ω. Suppose f, g agree on a non-empty subset of Ω, and this
subset has an accumulation point. Then f ≡ g on Ω.
• A consequence to this is that, non-trivial holomorphic functions have
isolated zeros.
Principle of analytical continuation
Let f : Ω → C be analytic on Ω and z0 ∈ Ω with B(z0, r) ⊆ Ω. Let γ be a
closed curve in Ω {z0} that is homotopic to a reparametrization of γ1 where
γ1(t) = z0 + reit
for t ∈ [0, 2π]. Then,
f(n)
(z0) =
n!
2πi
ˆ
γ
f(z)
(z − z0)n+1
dz
Cauchy estimates: If |f(z)| ≤ M ∀z ∈ γ([0, 2π]) then, ∀n ∈ N, then we have
|f(n)
(z0)| ≤ Mn!
rn
Higher-order Cauchy integral formula
Let f be a entire function which is bounded. Then f is a constant function.
Liouville’s Theorem
Let p(z) = a0+a1z+· · ·+anzn
be a non constant polynomial s.t. ai ∈ C, an 6= 0.
Then ∃z1, z2, . . . , zn s.t. p(z) = an(z − z1) . . . (z − zn).
Fundamental Theorem of Algebra
Let f : Ω → C be a continuous function such that,
´
γ
f(z) dz = 0, ∀ closed
polygonal paths γ ∈ Ω. Then f is holomorphic on Ω.
Morera’s Theorem
Let fn : Ω → C be a holomorphic on Ω, ∀n ∈ N s.t. fn converges uniformly on
compact sets to f. Then f is holomorphic.
Uniform limit of holomorphic functions
Let γ : [a, b] → C be a closed curve and let z0 be a point not in the image of
γ. Then the winding number of γ around z0 is
Wγ(z0) =
1
2πi
ˆ
γ
dz
z − z0
• Winding number is invariant over homotopy.
• Let z0 be a point not in the image of γ then ∃r 0 s.t. for z ∈
B(z0, r), Wγ(z0) = Wγ(z)
• The winding number is always an integer.
• The winding number is locally constant.
Generalized Cauchy Integral formula: Let f : Ω → C be holomorphic on
Ω and γ : [a, b] → Ω be a closed curve which is null homotopic. Then for z0
not in the image of γ,
f(z0)Wγ(z0) =
1
2πi
ˆ
γ
f(z)
(z − z0)
dz
Winding number

3. • f : Ω → C be holomorphic on Ω. Then G : Ω × Ω → C given by
G(z, w) =
f(z)−f(w)
z−w when z 6= w
f0
(z) when z = w
then G is continuous.
• Let f : Ω → C be holomorphic on some open set. Suppose z0 ∈ Ω s.t.
f0
(z0) 6= 0. Then ∃ a neighbourhood U of z0 ∈ Ω s.t. f restricted to U
is injective. And V = f(U) is an open set and the inverse g : V → U of
f is holomorphic.
• Let f : Ω → C be a non-constant holomorphic function on open, con-
nected set Ω. Let z0 ∈ Ω and w0 = f(z0). Then ∃ a neighbourhood U of
z0 and bijective holomorphic function ϕ on U s.t. f(z) = w0 + (ϕ(z))m
for z ∈ U and some integer m 0. And ϕ maps U unto B(0, r) for some
r 0.
Open Mapping Theorem: Let f : Ω → C be a non-constant holomorphic
function on open connected set Ω, then f(Ω) is an open set.
Open Mapping Theorem
Let Ω be a open connected set which is symmetric w.r.t R. Then define the
following,
• Ω+ = {z ∈ Ω | =(z) 0}
• Ω− = {z ∈ Ω | =(z) 0}
• I = {z ∈ Ω | =(z) = 0}
Schwarz reflection principle: Let Ω be defined as above. Then if f :
Ω+
S
I → C which is continuous on Ω+
S
I and holomorphic on Ω+. Suppose
for f(x) ∈ R, ∀x ∈ I then there exists g : Ω → C holomorphic on Ω s.t.
g(z) = f(z) for z ∈ Ω+
S
I
Schwarz reflection principle
• Isolated singularity: If f is holomorphic on B(z0, R) {z0} for some
R 0 then z0 is called an isolated singularity.
• Removable singularity: Let z0 be an isolated singularity of a holomor-
phic function f as defined above. It is called removable if there exists
holomorphic function g on B(z0, R) s.t. g(z) = f(z) on B(z0, R) {z0}.
• Riemann removable singularity theorem: Let z0 be an isolated
singularity of a function f, then z0 is a removable singularity if and only
if f is locally bounded around z0.
• Pole: If z0 is an isolated singularity as defined above and if lim
z→z0
|f(z)| =
∞ then z0 is called a pole of f.
• Essential singularity: A singularity that is neither removable nor a
pole.
Singularity of a holomorphic function
Let zn be a function defined for n =, 0, ±1, ±2, · · · , then it is doubly infinite.
• A doubly infinite series converges if
P∞
n=0 an and
P∞
n=1 a=n both con-
verge.
• Splitting up the series in similar manners you can define absolute and
uniform convergence.
Doubly infinite series
An annulus A(z0, R1, R2) around a point z0 for 0 ≤ R1 ≤ R2 is the set of all
z ∈ C s.t. R1 ≤ |z − z0| ≤ R2.
Annulus
Let f be a function holomorphic on A(z0, R1, R2), then there exists an ∈ C
for n ∈ Z s.t.
f(z) =
∞
X
n=−∞
an(z − z0)n
where the doubly infinite series converges absolutely and uniformly in some
A(z0, r1, r2) when R1 r1 r2 R2.
an =
1
2πi
ˆ
γ
f(z)
(z − z0)n+1
dz
where γ(z) = z0 + reit
for t ∈ [0, 2π] and R1 r R2.
Important results
• f has a removable singularity at z0 ⇐⇒ an = 0 for n 0 in the Laurent
series expansion of f
• f has a pole at z0 of order m ⇐⇒ an = 0 for n −m in the Laurent
series expansion of f.
• f has a essential singularity at z0 ⇐⇒ an 6= 0 for infinitely many
negative integers n.
Laurent series expansion
Let z0 be an essential singularity of f then given α ∈ C, there exists a
sequence zn ∈ B(z0, R) {z0} s.t. zn → z0 and f(zn) → α.
• Alternatively, f approaches any given value arbitrarily closely in any
neighborhood of an essential singularity.
Casorati–Weierstrass theorem
Let Ω be a open connected subset of C and let S ⊂ Ω. Let f : Ω S → C be
holomorphic on Ω. We say that f is a meromorphic function on Ω if,
• S is a discrete set.
• f either has removable singularities or poles at point of S.
Meromorphic functions
Let M(Ω) denote the equivalence classes of meromorphic functions over Ω.
• We say that two meromorphic functions f : Ω S1 and g : Ω S2 are
equivalent if f(z) = g(z) on Ω (S1
S
S2).
• For f, g ∈ M(Ω), define f + g to be the equivalence class of (f + g) :
Ω (S1
S
S2)
• Similarly, fg is the equivalence class of fg : Ω (S1
S
S2).
The space of all meromorphic functions is a field.
Operations on meromorphic functions
The order of a meromorphic function is defined as follows,
• If z0 ∈ S is a removable singularity then the order of f at z0 is the order
of the zero at z0 of f, i.e., f(z) = (z − z0)m
g(z) then m is the order.
• If z0 ∈ S is a pole and the pole is of order m then order of f at z0 is −m.
• If f ≡ 0 then Ordz0
= ∞.
• Ordz0
(f + g) ≥ min(Ordz0
(f), Ordz0
(g))
• Ordz0 (fg) = Ordz0 (f) + Ordz0 (g)
Order of meromorphic functions
Residue of a function: Let f : Ω S → C be a holomorphic function, where
Ω is an open set and S is a discrete subset of Ω. Then for z0 ∈ S, let r 0
be s.t. B(z0, r) ⊆ Ω and B(z0, r) = {z0}. Then in B(z0, r) {z0}, consider the
Laurent series expansion of f given by f(z) =
P∞
n=−∞ an(z − z0)n
. We define
the residue of f at z0 to be Res(f, z0) = a−1.
1. If z0 is a removable singularity then Res(z0) = 0.
2. If z0 is a pole of order m then (z − z0)m
f(z) = g(z), where g(z) 6= 0 on
B(z0, r) {z0}then, Res(z0) = am−1 = g(m−1)
(z0)
(m−1)! .
Residue of a function
Let Ω be an open connected subset of C and S be a finite subset of Ω and let
f : Ω S → C be a holomorphic function. Let γ be a null homotopic closed
curve on Ω. Then,
1
2πi
ˆ
γ
f(z) dz =
k
X
j=1
Wγ(zj)Res(f, zj)
where S = {z1, · · · , zk} and Wγ is the winding number.
Residue theorem
For a holomorphic function f : Ω → C. Define the log derivative of f to be the
meromorphic function f0
(z)
f(z) .
1. (fg)0
fg = f0
f + g0
g
2. (f/g)0
(f/g) = f0
f − g0
g
3. When f has a pole of order m at z0 then for f(z) = g(z)
(z−z0)m the log
derivative of f is g0
(z)
g(z) − m
(z−z0)
Log derivative
Let f : ΩS → C be a meromorphic function s.t. f has zeros of order d1, . . . , dn
at z1, . . . zn after removing the removable singularities. And f has poles of or-
der e1, . . . , em at points w1, . . . , wm. Let γ be a closed curve which is null
homotopic in Ω s.t. the zeros and poles don’t lie in the image of γ. Then,
1
2πi
ˆ
γ
f0
(z)
f(z)
dz =
n
X
i=0
diWγ(zi) −
m
X
j=1
ejWγ(wj)
Argument principle
Let γ be a closed curve which is null homotopic in Ω. Let f, g be functions
holomorphic in Ω and |g(z)| |f(z)| on γ then f and f + g have the same
number of zeros counting multiplicities on the interior of H([0, 1]×[a, b]) where
H is the null homotopy from γ to a constant path.
Rouche’s theorem
Let Ω be an open connected subset of C{0}. Define a branch of the logarithm
on Ω as a function f : Ω → C s.t. exp(f(z)) = z, ∀z ∈ Ω.
For Ω = C {(x) ≤ 0} define the standard branch to be
Log(z) = ln |z| + iArg(z)
As defined above Log(z) is holomorphic on Ω.
Branch of the complex logarithm

4. Let D denote the open unit disc. Let f : D → D be a holomoprhic function s.t.
f(0) = 0. Then,
|f(z)| ≤ |z|, ∀z ∈ D, and |f0
(z)| ≤ 1
Also, if |f(z)| = |z| for some z ∈ D or if |f0
(0)| = 1 then ∃λ ∈ C, |λ| = 1 s.t.
f(z) = λz.
Schwarz lemma
A function f : Ω → Ω is an automorphism if f is holomorphic and has a
holomorphic inverse.
Automorphism
Define a function ϕα : D → C defined as ϕα(z) = z−α
1−αz .
Let f : D → D be an automorphism. Then there exists α ∈ D and λ ∈ ∂D s.t.
f(z) = λϕα(z)
Automorphisms of the unit disc
Let Ω = {z ∈ Ω : a (z) b}. Let f : Ω → C, s.t. f is continuous on Ω and
holomorphic on Ω. Suppose for some z = x + iy, we have |f(z)| B and let
M(x) = sup{|f(x + iy)| : −∞ y ∞}. Then,
M(x)b−a
≤ M(a)b−x
M(b)x−a
And further
|f(z)| ≤ M(x) ≤ max{M(a), M(b)} = sup
z∈∂Ω
|f(z)|
Phragmén–Lindelöf method
First define ρ(z, w) = z−w
1−wz for z, w ∈ D. Let f : D → D be holomorphic.
Then,
ρ(f(z), f(w)) ≤ ρ(z, w) ∀z, w ∈ D
and,
|f0
(z)|
1 − |f(z)|2
≤
1
1 − |z|2
∀z ∈ D
Schwarz-Pick theorem
Let X, Y, Z be open subsets of C and let f : Y → X and g : Z → X be
continuous maps. Then we say, a map e
g : Z → Y is a lift of g w.r.t. f if f◦e
g = g.
Uniqueness of lifts: Let X, Y, Z be open connected subsets of C and let
f : Y → X be a local homeomorphism. Let g : Z → X be a continuous map.
Let e
g1 and e
g2 be lifts of g w.r.t. f and suppose they are equal at some point
in Z. Then e
g1 ≡ e
g2.
• Let f : Y → X be a holomorphic map s.t. f0
(y) 6= 0 on Y . Let g : Z → X
be a holomorphic map s.t. e
g : Z → Y is a lift of g w.r.t. f. Then e
g is
holomorphic.
• Let X, Y be open subsets of C let, f : Y → X be a local homeomor-
phism. Let γ0, γ1 be curves in X from z1 to z2 which are homotopic.
Suppose that for every s ∈ [0, 1], we can lift γs(t) = H(s, t) to a path
e
γs : [a, b] → Y w.r.t. f s.t. e
γs(a) = e
z1, ∀s ∈ [0, 1]. Then e
γ0, e
γ1 are
homotopic in Y .
Lifting of maps
Let X, Y be open subsets of C. We say that a continuous map f : Y → X
is a covering map if given x ∈ X there exists a neighbourhood U of X and
open sets {Vα}α∈A in Y s.t. f−1
(U) =
`
α∈A Vα (disjoint union of Vα) and
f|Vα : Vα → U is a homeomorphism. Then Y is called a cover of X.
• Let f : Y → X be a covering map and γ[a, b] → X be a curve from
x0 to x1 in X. Suppose y0 ∈ f−1
({x0}). Then there exists a unique lift
e
γ[a, b] → Y of γ w.r.t. f s.t. e
γ(a) = y0.
• For connected X let f : Y → X be a covering map. Suppose x0, x1 ∈ X.
Then the cardinality of f−1
(x0) is the same as the cardinality of f−1
(x).
• For open subsets X, Y of C let, f : Y → X be a covering map from Y to
X. Let Z be an open connected subset of C, which is simply connected
and locally connected. Suppose g : Z → X is a continuous map. Then
given z0 ∈ C and y0 ∈ Y s.t. g(z0) = f(y0), then there exists a unique
lift e
g : Z → Y of g w.r.t f.
• Let Ω be a simply connected, locally connected, open connected subset
of C and g : Ω → C∗
be a holomorphic map. Then there exists a lift
e
g : Ω → C s.t. exp(e
g) = g.
Covering spaces
• For f : D → C s.t. f(0) = 0, f0
(0) = 1 and |f(z)| ≤ M ∀z ∈ D. Then
B(0, 1
6M ) ⊆ f(D).
• Let f : B(0, R) → C be holomorphic s.t. f(0) = 0, f0
(0) = µ for some
µ 0 and f|(z)| ≤ M ∀z ∈ B(0, R). Then, B(0, R2
µ2
6M ) ⊆ f(B(0, R)).
Bloch’s theorem: Let Ω be an open connected subset of C s.t. D ⊂ Ω.
Letf : Ω → C s.t. f(0) = 0, f0
(0) = 1. Then there exists a ball B0
contained
in D s.t. f|B0 is injective and B(0, 1
72 ) ⊆ f(B0
) ⊆ f(D).
Bloch’s theorem
• Let Ω be an open connected subset of C which is simply connected. Let
f : Ω → C which omits 0 and 1. Then there exists a holomorphic function
g : Ω → C s.t. f(z) = − exp(πi cosh(2g(z)))
• The function g as defined above doesn’t contain any disk of radius 1.
Little Picard’s theorem: If f is an entire function which omits two points,
then f is a constant function.
Little Picard’s theorem