1Analytic functions

IB Complex Methods

1.4 Multi-valued functions
For
z
=
r
, we define
log z
=
log r
+
. 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
z1
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
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
2π
to
r
α
0
. So
z
α
is not continuous if
e
2π
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
z1
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,
1ζ
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
1ζ
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|
+
. Just below it, at
z
=
x
+
i
0
,
we have log z = log |x| . 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.
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
and
z
1 =
r
1
e
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-
f(z) = log
z 1
z + 1
.
Writing z + 1 = re
and z 1 = r
1
e
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:
11
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.