1Analytic functions

IB Complex Methods

1.4 Multi-valued functions

For

z

=

r

iθ

, we define

log z

=

log r

+

iθ

. There are infinitely many values of

log z, for every choice of θ. For example,

log i =

πi

2

or

5πi

2

or −

3πi

2

or ··· .

This is fine, right? Functions can be multi-valued. Nothing’s wrong.

Well, when we write down an expression, it’d better be well-defined. So we

really should find some way to deal with this.

This section is really more subtle than it sounds like. It turns out it is non-

trivial to deal with these multi-valued functions. We can’t just, say, randomly

require

θ

to be in, say,

(0, 2π]

, or else we will have some continuity problems, as

we will later see.

Branch points

Consider the three curves shown in the diagram.

C

3

C

1

C

2

In

C

1

, we could always choose

θ

to be always in the range

0,

π

2

, and then

log z

would be continuous and single-valued going round C

1

.

On

C

2

, we could choose

θ ∈

π

2

,

3π

2

and

log z

would again be continuous

and single-valued.

However, this doesn’t work for

C

3

. Since this encircles the origin, there is no

such choice. Whatever we do,

log z

cannot be made continuous and single-valued

around

C

3

. It must either “jump” somewhere, or the value has to increase by

2πi every time we go round the circle, i.e. the function is multi-valued.

We now define what a branch point is. In this case, it is the origin, since

that is where all our problems occur.

Definition

(Branch point)

.

A branch point of a function is a point which is

impossible to encircle with a curve on which the function is both continuous and

single-valued. The function is said to have a branch point singularity there.

Example.

(i) log(z − a) has a branch point at z = a.

(ii) log

z−1

z+1

= log(z − 1) − log(z + 1) has two branch points at ±1.

(iii) z

α

=

r

α

e

iαθ

has a branch point at the origin as well for

α 6∈ Z

— consider

a circle of radius of

r

0

centered at 0, and wlog that we start at

θ

= 0 and

go once round anticlockwise. Just as before,

θ

must vary continuous to

ensure continuity of

e

iαθ

. So as we get back almost to where we started,

θ

will approach 2

π

, and there will be a jump in

θ

from 2

π

back to 0. So

there will be a jump in

z

α

from

r

α

0

e

2πiα

to

r

α

0

. So

z

α

is not continuous if

e

2πiα

6= 1, i.e. α is not an integer.

(iv) log z

also has a branch point at

∞

. Recall that to investigate the properties

of a function

f

(

z

) at infinity, we investigate the property of

f

1

z

at zero.

If

ζ

=

1

z

, then

log z

=

−log ζ

, which has a branch point at

ζ

= 0. Similarly,

z

α

has a branch point at ∞ for α 6∈ Z.

(v)

The function

log

z−1

z+1

does not have a branch point at infinity, since if

ζ =

1

z

, then

log

z −1

z + 1

= log

1 − ζ

1 + ζ

.

For

ζ

close to zero,

1−ζ

1+ζ

remains close to 1, and therefore well away from

the branch point of

log

at the origin. So we can encircle

ζ

= 0 without

log

1−ζ

1+ζ

being discontinuous.

So we’ve identified the points where the functions have problems. How do

we deal with these problems?

Branch cuts

If we wish to make

log z

continuous and single valued, therefore, we must stop

any curve from encircling the origin. We do this by introducing a branch cut

from −∞ on the real axis to the origin. No curve is allowed to cross this cut.

z

θ

Once we’ve decided where our branch cut is, we can use it to fix on values of

θ

lying in the range (

−π, π

], and we have defined a branch of

log z

. This branch

is single-valued and continuous on any curve

C

that does not cross the cut.

This branch is in fact analytic everywhere, with

d

dz

log z

=

1

z

, except on the

non-positive real axis, where it is not even continuous.

Note that a branch cut is the squiggly line, while a branch is a particular

choice of the value of log z.

The cut described above is the canonical (i.e. standard) branch cut for

log z

.

The resulting value of log z is called the principal value of the logarithm.

What are the values of

log z

just above and just below the branch cut?

Consider a point on the negative real axis,

z

=

x <

0. Just above the cut, at

z

=

x

+

i

0

+

, we have

θ

=

π

. So

log z

=

log |x|

+

iπ

. Just below it, at

z

=

x

+

i

0

−

,

we have log z = log |x| − iπ. Hence we have a discontinuity of 2πi.

We have picked an arbitrary branch cut and branch. We can pick other

branch cuts or branches. Even with the same branch cut, we can still have a

different branch — we can instead require

θ

to fall in (

π,

3

π

]. Of course, we can

also pick other branch cuts, e.g. the non-negative imaginary axis. Any cut that

stops curves wrapping around the branch point will do.

Here we can choose θ ∈

−

3π

2

,

π

2

. We can also pick a branch cut like this:

The exact choice of

θ

is more difficult to write down, but this is an equally valid

cut, since it stops curves from encircling the origin.

Exactly the same considerations (and possible branch cuts) apply for

z

α

(for

α 6∈ Z).

In practice, whenever a problem requires the use of a branch, it is important

to specify it clearly. This can be done in two ways:

(i) Define the function and parameter range explicitly, e.g.

log z = log |z| + i arg z, arg z ∈ (−π, π].

(ii)

Specify the location of the branch cut and give the value of the required

branch at a single point not on the cut. The values everywhere else are

then defined uniquely by continuity. For example, we have

log z

with a

branch cut along

R

≤0

and

log

1 = 0. Of course, we could have defined

log 1 = 2πi as well, and this would correspond to picking arg z ∈ (π, 3π].

Either way can be used, but it must be done properly.

Riemann surfaces*

Instead of this brutal way of introducing a cut and forbidding crossing, Riemann

imagined different branches as separate copies of

C

, all stacked on top of each

other but each one joined to the next at the branch cut. This structure is a

Riemann surface.

C

C

C

C

C

The idea is that traditionally, we are not allowed to cross branch cuts. Here,

when we cross a branch cut, we will move to a different copy of

C

, and this

corresponds to a different branch of our function.

We will not say any more about this — there is a whole Part II course

devoted to these, uncreatively named IID Riemann Surfaces.

Multiple branch cuts

When there is more than one branch point, we may need more than one branch

cut. For

f(z) = (z(z −1))

1

3

,

there are two branch points, at 0 and 1. So we need two branch cuts. A possibility

is shown below. Then no curve can wrap around either 0 or 1.

10

z

r

r

1

θ

θ

1

For any

z

, we write

z

=

re

iθ

and

z −

1 =

r

1

e

iθ

1

with

θ ∈

(

−π, π

] and

θ

1

∈

[0

,

2

π

),

and define

f(z) =

3

√

rr

1

e

i(θ+θ

1

)/3

.

This is continuous so long as we don’t cross either branch cut. This is all and

simple.

However, sometimes, we need fewer branch cuts than we might think. Con-

sider instead the function

f(z) = log

z −1

z + 1

.

Writing z + 1 = re

iθ

and z − 1 = r

1

e

iθ

1

, we can write this as

f(z) = log(z − 1) − log(z + 1)

= log(r

1

/r) + i(θ

1

− θ).

This has branch points at

±

1. We can, of course, pick our branch cut as above.

However, notice that these two cuts also make it impossible for

z

to “wind

around

∞

” (e.g. moving around a circle of arbitrarily large radius). Yet

∞

is not

a branch point, and we don’t have to make this unnecessary restriction. Instead,

we can use the following branch cut:

1−1

z

r

r

1

θ

θ

1

Drawing this branch cut is not hard. However, picking the values of

θ, θ

1

is

more tricky. What we really want to pick is

θ, θ

1

∈

[0

,

2

π

). This might not look

intuitive at first, but we will shortly see why this is the right choice.

Suppose that we are unlawful and cross the branch cut. Then the value of

θ

passes through the branch cut, while the value of

θ

1

varies smoothly. So the

value of

f

(

z

) jumps. This is expected since we have a branch cut there. If we

pass through the negative real axis on the left of the branch cut, then nothing

happens, since θ = θ

1

= π are not at a point of discontinuity.

The interesting part is when we pass through the positive real axis on the

right of branch cut. When we do this, both

θ

and

θ

1

jump by 2

π

. However, this

does not induce a discontinuity in

f

(

z

), since

f

(

z

) depends on the difference

θ

1

− θ, which has not experienced a jump.