1Analytic functions

IB Complex Methods

1.2 Complex differentiation

Recall the definition of differentiation for a real function f(x):

f

0

(x) = lim

δx→0

f(x + δx) − f(x)

δx

.

It is implicit that the limit must be the same whichever direction we approach

from. For example, consider

|x|

at

x

= 0. If we approach from the right, i.e.

δx →

0

+

, then the limit is +1, whereas from the left, i.e.

δx →

0

−

, the limit is

−

1. Because these limits are different, we say that

|x|

is not differentiable at

the origin.

This is obvious and we already know that, but for complex differentiation,

this issue is much more important, since there are many more directions. We

now extend the definition of differentiation to complex number:

Definition

(Complex differentiable function)

.

A complex differentiable function

f : C → C is differentiable at z if

f

0

(z) = lim

δz→0

f(z + δz) − f(z)

δz

exists (and is therefore independent of the direction of approach — but now

there are infinitely many possible directions).

This is the same definition as that for a real function. Often, we are not

interested in functions that are differentiable at a point, since this might allow

some rather exotic functions we do not want to consider. Instead, we want the

function to be differentiable near the point.

Definition

(Analytic function)

.

We say

f

is analytic at a point

z

if there

exists a neighbourhood of

z

throughout which

f

0

exists. The terms regular and

holomorphic are also used.

Definition

(Entire function)

.

A complex function is entire if it is analytic

throughout C.

The property of analyticity is in fact a surprisingly strong one! For example,

two consequences are:

(i)

If a function is analytic, then it is differentiable infinitely many times. This

is very very false for real functions. There are real functions differentiable

N

times, but no more (e.g. by taking a non-differentiable function and

integrating it N times).

(ii) A bounded entire function must be a constant.

There are many more interesting properties, but these are sufficient to show us

that complex differentiation is very different from real differentiation.

The Cauchy-Riemann equations

We already know well how to differentiate real functions. Can we use this to

determine whether certain complex functions are differentiable? For example is

the function

f

(

x

+

iy

) =

cos x

+

i sin y

differentiable? In general, given a complex

function

f(z) = u(x, y) + iv(x, y),

where

z

=

x

+

iy

are

u, v

are real functions, is there an easy criterion to determine

whether f is differentiable?

We suppose that

f

is differentiable at

z

. We may take

δz

in any direction

we like. First, we take it to be real, with δz = δx. Then

f

0

(z) = lim

δx→0

f(z + δx) − f (z)

δx

= lim

δx→0

u(x + δx, y) + iv(x + δx, y) − (u(x, y) + iv(x, y))

δx

=

∂u

∂x

+ i

∂v

∂x

.

What this says is something entirely obvious — since we are allowed to take the

limit in any direction, we can take it in the

x

direction, and we get the corre-

sponding partial derivative. This is a completely uninteresting point. Instead,

let’s do the really fascinating thing of taking the limit in the y direction!

Let δz = iδy. Then we can compute

f

0

(z) = lim

δy→0

f(z + iδy) − f(z)

iδy

= lim

δy→0

u(x, y + δy) + iv(x, y + δy) − (u(x, y) + iv(x, y))

iδy

=

∂v

∂y

− i

∂u

∂y

.

By the definition of differentiability, the two results for

f

0

(

z

) must agree! So we

must have

∂u

∂x

+ i

∂v

∂x

=

∂v

∂y

− i

∂u

∂y

.

Taking the real and imaginary components, we get

Proposition

(Cauchy-Riemann equations)

.

If

f

=

u

+

iv

is differentiable, then

∂u

∂x

=

∂v

∂y

,

∂u

∂y

= −

∂v

∂x

.

Is the converse true? If these equations hold, does it follow that

f

is differ-

entiable? This is not always true. This holds only if

u

and

v

themselves are

differentiable, which is a stronger condition that the partial derivatives exist, as

you may have learnt from IB Analysis II. In particular, this holds if the partial

derivatives u

x

, u

y

, v

x

, v

y

are continuous (which implies differentiability). So

Proposition.

Given a complex function

f

=

u

+

iv

, if

u

and

v

are real differen-

tiable at a point z and

∂u

∂x

=

∂v

∂y

,

∂u

∂y

= −

∂v

∂x

,

then f is differentiable at z.

We will not prove this — proofs are for IB Complex Analysis.

Examples of analytic functions

Example.

(i) f

(

z

) =

z

is entire, i.e. differentiable everywhere. Here

u

=

x, v

=

y

. Then

the Cauchy-Riemann equations are satisfied everywhere, since

∂u

∂x

=

∂v

∂y

= 1,

∂u

∂y

= −

∂v

∂x

= 0,

and these are clearly continuous. Alternatively, we can prove this directly

from the definition.

(ii) f(z) = e

z

= e

x

(cos y + i sin y) is entire since

∂u

∂x

= e

x

cos y =

∂v

∂y

,

∂u

∂y

= −e

x

sin y = −

∂v

∂x

.

The derivative is

f

0

(z) =

∂u

∂x

+ i

∂v

∂x

= e

x

cos y + ie

x

sin y = e

z

,

as expected.

(iii) f

(

z

) =

z

n

for

n ∈ N

is entire. This is less straightforward to check. Writing

z = r(cos θ + i sin θ), we obtain

u = r

n

cos nθ, v = r

n

sin nθ.

We can check the Cauchy-Riemann equation using the chain rule, writing

r

=

p

x

2

= y

2

and

tan θ

=

y

x

. This takes quite a while, and it’s not worth

your time. But if you really do so, you will find the derivative to be

nz

n−1

.

(iv)

Any rational function, i.e.

f

(

z

) =

P (z)

Q(z)

where

P, Q

are polynomials, is

analytic except at points where

Q

(

z

) = 0 (where it is not even defined).

For instance,

f(z) =

z

z

2

+ 1

is analytic except at ±i.

(v)

Many standard functions can be extended naturally to complex functions

and obey the usual rules for their derivatives. For example,

– sin z

=

e

iz

−e

−iz

2i

is differentiable with derivative

cos z

=

e

iz

+e

−iz

2

. We

can also write

sin z = sin(x + iy)

= sin x cos iy + cos x sin iy

= sin x cosh y + i cos x sinh y,

which is sometimes convenient.

–

Similarly

cos z, sinh z, cosh z

etc. differentiate to what we expect them

to differentiate to.

– log z = log |z| + i arg z has derivative

1

z

.

–

The product rule, quotient rule and chain rule hold in exactly the

same way, which allows us to prove (iii) and (iv) easily.

Examples of non-analytic functions

Example.

(i) Let f(z) = Re z. This has u = x, v = 0. But

∂u

∂x

= 1 6= 0 =

∂v

∂y

.

So Re z is nowhere analytic.

(ii)

Consider

f

(

z

) =

|z|

. This has

u

=

p

x

2

+ y

2

, v

= 0. This is thus nowhere

analytic.

(iii)

The complex conjugate

f

(

z

) =

¯z

=

z

∗

=

x − iy

has

u

=

x, v

=

−y

. So the

Cauchy-Riemann equations don’t hold. Hence this is nowhere analytic.

We could have deduced (ii) from this — if

|z|

were analytic, then so would

|z|

2

, and hence ¯z =

|z|

2

z

also has to be analytic, which is not true.

(iv)

We have to be a bit more careful with

f

(

z

) =

|z|

2

=

x

2

+

y

2

. The

Cauchy-Riemann equations are satisfied only at the origin. So

f

is only

differentiable at

z

= 0. However, it is not analytic since there is no

neighbourhood of 0 throughout which f is differentiable.