1Complex differentiation
IB Complex Analysis
1.1 Differentiation
We start with some definitions. As mentioned in the introduction, Liouville’s
theorem says functions defined on the whole of
C
are often not that interesting.
Hence, we would like to work with some subsets of
C
instead. As in real analysis,
for differentiability to be well-defined, we would want a function to be defined
on an open set, so that we can see how
f
:
U → C
varies as we approach a point
z
0
∈ U from all different directions.
Definition
(Open subset)
.
A subset
U ⊆ C
is open if for any
x ∈ U
, there is
some ε > 0 such that the open ball B
ε
(x) = B(x; ε) ⊆ U.
The notation used for the open ball varies form time to time, even within
the same sentence. For example, instead of putting
ε
as the subscript, we could
put
x
as the subscript and
ε
inside the brackets. Hopefully, it will be clear from
context.
This is good, but we also want to rule out some silly cases, such as functions
defined on subsets that look like this:
This would violate results such as functions with zero derivative must be constant.
Hence, we would require our subset to be connected. This means for any two
points in the set, we can find a path joining them. A path can be formally
defined as a function γ : [0, 1] → C, with start point γ(0) and end point γ(1).
Definition
(Path-connected subset)
.
A subset
U ⊆ C
is path-connected if for
any
x, y ∈ U
, there is some
γ
: [0
,
1]
→ U
continuous such that
γ
(0) =
x
and
γ(1) = y.
Together, these define what it means to be a domain. These are (usually)
things that will be the domains of our functions.
Definition
(Domain)
.
A domain is a non-empty open path-connected subset
of C.
With this, we can define what it means to be differentiable at a point. This
is, in fact, exactly the same definition as that for real functions.
Definition
(Differentiable function)
.
Let
U ⊆ C
be a domain and
f
:
U → C
be a function. We say f is differentiable at w ∈ U if
f
0
(w) = lim
z→w
f(z) −f (w)
z − w
exists.
Here we implicitly require that the limit does not depend on which direction
we approach
w
from. This requirement is also present for real differentiability, but
there are just two directions we can approach
w
from — the positive direction
and the negative direction. For complex analysis, there are infinitely many
directions to choose from, and it turns out this is a very strong condition to
impose.
Complex differentiability at a point
w
is not too interesting. Instead, what we
want is a slightly stronger condition that the function is complex differentiable
in a neighbourhood of w.
Definition
(Analytic/holomorphic function)
.
A function
f
is analytic or holo-
morphic at
w ∈ U
if
f
is differentiable on an open neighbourhood
B
(
w, ε
) of
w
(for some ε).
Definition
(Entire function)
.
If
f
:
C → C
is defined on all of
C
and is
holomorphic on C, then f is said to be entire.
It is not universally agreed what the words analytic and holomorphic should
mean. Some people take one of these word to mean instead that the function
has a (and is given by the) Taylor series, and then take many pages to prove
that these two notions are indeed the same. But since they are the same, we
shall just opt for the simpler definition.
The goal of the course is to develop the rich theory of these complex dif-
ferentiable functions and see how we can integrate them along continuously
differentiable (C
1
) paths in the complex plane.
Before we try to achieve our lofty goals, we first want to figure out when
a function is differentiable. Sure we can do this by checking the definition
directly, but this quickly becomes cumbersome for more complicated functions.
Instead, we would want to see if we can relate complex differentiability to real
differentiability, since we know how to differentiate real functions.
Given
f
:
U → C
, we can write it as
f
=
u
+
iv
, where
u, v
:
U → R
are
real-valued functions. We can further view
u
and
v
as real-valued functions of
two real variables, instead of one complex variable.
Then from IB Analysis II, we know this function
u
:
U → R
is differentiable
(as a real function) at a point (
c, d
)
∈ U
, with derivative
Du|
(c,d)
= (
λ, µ
), if and
only if
u(x, y) −u(c, d) − (λ(x − c) + µ(y − d))
k(x, y) − (c, d)k
→ 0 as (x, y) → (c, d).
This allows us to come up with a nice criterion for when a complex function is
differentiable.
Proposition.
Let
f
be defined on an open set
U ⊆ C
. Let
w
=
c
+
id ∈ U
and
write
f
=
u
+
iv
. Then
f
is complex differentiable at
w
if and only if
u
and
v
,
viewed as a real function of two real variables, are differentiable at (c, d), and
u
x
= v
y
,
u
y
= −v
x
.
These equations are the Cauchy–Riemann equations. In this case, we have
f
0
(w) = u
x
(c, d) + iv
x
(c, d) = v
y
(c, d) − iu
y
(c, d).
Proof. By definition, f is differentiable at w with f
0
(w) = p + iq if and only if
lim
z→w
f(z) − f (w) − (p + iq)(z − w)
z − w
= 0. (†)
If z = x + iy, then
(p + iq)(z − w) = p(x − c) − q(y − d) + i(q(x − c) + p(y − d)).
So, breaking into real and imaginary parts, we know (†) holds if and only if
lim
(x,y)→(c,d)
u(x, y) − u(c, d) − (p(x − c) − q(y − d))
p
(x − c)
2
+ (y − d)
2
= 0
and
lim
(x,y)→(c,d)
v(x, y) − v(c, d) − (q(x − c) + p(y − d))
p
(x − c)
2
+ (y − d)
2
= 0.
Comparing this to the definition of the differentiability of a real-valued function,
we see this holds exactly if u and v are differentiable at (c, d) with
Du|
(c,d)
= (p, −q), Dv|
(c,d)
= (q, p).
A standard warning is given that
f
:
U → C
can be written as
f
=
u
+
iv
,
where
u
x
=
v
y
and
u
y
=
−v
x
at (
c, d
)
∈ U
, we cannot conclude that
f
is complex
differentiable at (
c, d
). These conditions only say the partial derivatives exist,
but this does not imply imply that
u
and
v
are differentiable, as required by the
proposition. However, if the partial derivatives exist and are continuous, then
by IB Analysis II we know they are differentiable.
Example.
(i)
The usual rules of differentiation (sum rule, product, rule, chain rule,
derivative of inverse) all hold for complex differentiable functions, with the
same proof as the real case.
(ii)
A polynomial
p
:
C → C
is entire. This can be checked directly from
definition, or using the product rule.
(iii)
A rational function
p(z)
q(z)
:
U → C
, where
U ⊆ C \ {z
:
q
(
z
) = 0
}
, is
holomorphic on any such U . Here p, q are polynomials.
(iv) f
(
z
) =
|z|
is not complex differentiable at any point of
C
. Indeed, we can
write this as f = u + iv, where
u(x, y) =
p
x
2
+ y
2
, v(x, y) = 0.
If (x, y) 6= (0, 0), then
u
x
=
x
p
x
2
+ y
2
, u
y
=
y
p
x
2
+ y
2
.
If we are not at the origin, then clearly we cannot have both vanish, but
the partials of
v
both vanish. Hence the Cauchy–Riemann equations do
not hold and it is not differentiable outside of the origin.
At the origin, we can compute directly that
f(h) − f(0)
h
=
|h|
h
.
This is, say, +1 for
h ∈ R
+
and
−
1 for
h ∈ R
−
. So the limit as
h →
0 does
not exist.