COMPLEX GEOMETRY Course notes

Chapter 1
COMPLEX ANALYSIS
1.1 Complex Analysis in one variable
Let U ⊆ C = R 2 be an open subset of the complex plane. We shall denote an element z ∈ C by z = x + iy,
where i = √ −1. An function f : U −→ C is holomorphic on U if it is complex differentiable at all points
of U, i.e.,
f ′ (z 0 ) = df
dz (z 0) = lim z→z0
f(z)−f(z 0)
z−z 0
exists for every z 0 ∈ U. We shall denote this by f ∈ O(U). If S ⊆ C is any subset, we shall say that f is
holomorphic on S (f ∈ O(S)) if f is holomorphic on a open neighbourhood of S.
If the function f is R-differentiable on U then ∂f ∂f
∂x
dx +
∂y makes sense and df(x, y) ∈ Hom R(T z=x+iy U, R 2 ).
Recall that
dz = dx + idy and dz = dx − idy
Using these expressions, we can write the differential df as
df = 1 2
( ) ( )
∂f
∂x − i ∂f
∂y
dz + 1 ∂f
2 ∂x + i ∂f
∂y
dz = ∂f ∂f
∂z
dz +
∂z dz
Notice the following relations
( )
∂f
∂z
= ∂f
∂z
and
( )
∂f
∂z
= ∂f
∂z
Recall that
df =
f is complex differentiable
⇐⇒
f is R-differentiable and ∂f
∂z
= 0 (Cauchy-Riemann condition)
⇐⇒
∂u ∂v
∂z
= −i
∂z
( )
⇐⇒
ux u y
is a rotation matrix up to a real scalar multiple (u
v x v x = v y and u y = −v x ).
y
1