Kusse B.R., Westwig E.A. Mathematical Physics: Applied Mathematics for Scientists and Engineers
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526
ADVANCED
TOPICS
IN
COMPLEX
ANALYSIS
which is a factor
of
-
1
different from its initial value.
This
shows
5
=
Q
is
a branch
point of
w
=
(z2
-
u2)'I2.
A
tour around the point
z
=
-a
produces a similar result,
so
the two points
z
=
+a
are both branch points
of
y.
We have already shown there
are
no
other branch points in the finite z-plane, but
there
is
the possibility that these two branch points
do
not
fom
a
pair with a single cut
between them. Instead, each of the branch points might have a separate cut extending
out to infinity. In
this
case, you can
visualize
each of the two branch points paired
with a different branch point at infinity.
An
easy way to determine
if
this
is
actually
happening in
this
case is to make a large tour that encloses both branch points, as
shown in Figure
13.19.
If,
at the end of
this
tour, we are on
a
different sheet from the
start, we must have crossed some cut(s) extending to infinity. After performing
this
big tour, notice both
rl
and
r2
have returned to their
initial
values, but
both
81
and
62
have increased by
2n-,
so
w
becomes:
(13.25)
which is exactly the initial value of
w.
We did not have to cross a cut,
so
the two
branch points at
g
=
+a
must
form
a
pair
with a single cut between them.
To
determine the complete
nature
of
the sheets we need to know the results of a
few
more
tours.
It
is
easy to show
that
touring around either of the branch points at
z
=
+a
twice returns to its initial value.
Also,
going around one of the branch
-
z-plane
-+-~-
-
z
-
a
\
'%
X
l
'
-a a,
Figure
13.19
Tour
Around
Both
Branch
Points
MULTIVALUED FUNCTIONS
527
2
Sheet 1
Sheet 2
Figure
13.20
Riemann Sheet
Connections
points once and then back around the other, in a figure-eight pattern, also returns
111
to
its initial value. These facts lead us to conclude there are two Riemann sheets in the
g-plane, and they are connected with a cut between them,
as
shown in Figure 13.20.
This figure shows the cut placed along a straight line between
z
=
?-+a.
As
we
mentioned earlier, the cut can be placed anywhere as long as it terminates at the
appropriate branch points. We could legitimately deform the cut, in successive steps,
as shown in Figure 13.21. The configuration of the cut at the bottom
of
this figure
makes it look like there might now be two branch cuts.
This
is not
the
case.
It
is
actually the same cut making a very large loop around the complex plane. With the
cut positioned as shown at the bottom of Figure
13.21,
a tour enclosing both branch
points crosses the branch cut twice. Because
this
is the same cut, however, the second
crossing
of
the cut returns us to the original sheet.
Example
13.6
As
a last example, consider the function
(13.26)
Although this looks very similar
to
the function in the previous example, it behaves
quite differently.
To
find the branch points, tours must be made in the z-plane. The
same phasors that were used in the previous problem, and shown in Figure 13.16,
can be used again here:
z
+
a
=
rl eiel
z
-
a
=
r2e'h.
With these phasors, the function
can
be
rewritten
(13.27)
(13.28)
In a single, counterclockwise lap around
g
=
a,
rl,
81
and
1-2
return to their initial
values. The angle
192,
however, increases
by
27~. For simplicity, take the initial values
of
81
and
02
to be zero, and the initial radial values to be
rl
=
rli
and
rz
=
rzi.
Then
528
-a
ADVANCED TOPICS IN COMPLEX ANALYSIS
a
-a
a
-a
I
a
X
----
-
,
-a
a
i
l
Figure
13.21
Deformation
of
the
Branch
Cut
at
the
start
of
this
tour,
but at
the
end
-
w=Jr,-Jr,.
(13.29)
(13.30)
MULTIVALUED FUNCTIONS
529
A
tour around
z
=
-a
produces a similar result,
so
both points at
z
=
ta
are branch
points. Single
loop
tours around any other point in the finite _z-plane will return to
its starting value,
so
there are no other branch points in the finite z-plane.
As
was the case in the previous example, we must check to see if there are any
branch points at infinity by making
a
tour around both
g
=
a
and
z
=
-a.
Start the
tour with both
8,
and
&
equal to zero. At the end of this tour,
11
and
rz
will return to
their initial values, but both
81
and
82
will increase by
27r.
Thus, at the start
of
this
tour
w=Jr,,+Jr,,.
while at the end
-
w=-Jlll-
Jr,,.
(13.31)
(
13.32)
Consequently, there must be two branch points at infinity, and the branch cuts that
start
at
z
=
a
and
z
=
-a
extend to infinity, as shown in Figure
13.22.
Unlike the
single cut, with the-ksleading appearance
of
two cuts, shown at the bottom
of
Figure
13.21,
these really are two distinct cuts extending out to infinity.
We can determine the total number
of
Riemann sheets and how they are connected
at the cuts by performing
a
few more tours, and watching how many different values
of
are generated at the same point of
g,
Consider the three different tours shown
in Figure
13.23.
These tours
all
start with the same value of
g.
This value, which we
will call
zi
=
ri
e”1,
is a pure real quantity, lying just above the cut with
8i
=
0
and
to the right of
z
=
a.
In terms of the phasor parameters, at the start
of
each
of
these
tours
w=Jr,+Jr,.
We already determined that at the end of the topmost tour
Y
-
z-plane
X
-
-a a
(13.33)
(13.34)
Figure
13.22
Branch
Points
and Branch
Cuts
for
=
(z_
-
a)’/*
+
(z_
+
530
ADVANCED TOPICS IN COMPLEX ANALYSIS
e
-a
-
z-plane
Iy
i
I
I
i-
I
I
,
a
\
1
/'
\
Y
Y
\
-a
\
Figure
13.23
Tours
to
Determine the Number
of
Riemann Sheets
which is different
from
the initial value,
so
at the end
of
this
tour we must be on a
second Riemann sheet. Now consider
the
tour shown in the middle
of
Figure
13.23.
After one lap around
this
path,
only
81
has changed
by
277,
and
so
-
w=-fi+fi.
(13.35)
531
MULTIVALUED FUNCTIONS
rhis
is a third value for
w,
different from both Equation
13.34
and
13.35,
so
at the
:nd
of
this tour, we must be on a third Riemann sheet.
A
single lap around the tour
shown on the bottom
of
Figure
13.23
changes both
O1
and
O2
by
2~.
At the end of
:his tour
g=-Jr,,-fi.
(13.36)
rhis is a fourth value for
g
and consequently requires a fourth Riemann sheet.
Any combination of these tours or multiple laps of a single tour will not generate
any
new values for
y,
and
so
we conclude that there are four Riemann sheets. Because
a
two-lap tour around either of the branch points at
z
=
+a
would return us to the
ariginal sheet, but a figure-eight pattern would not,
the
connection diagram must be
as
shown in Figure
13.24.
I
\
\
\
4
b
Figure
13.24
Riemann
Sheet
Connections
532
ADVANCED TOPICS IN COMPLEX ANALYSIS
A quicker way to see that there should
be
four Riemann sheets for
this
function is
to realize that the
2
property of the square root operation, i.e.,
-
w
=
?fit
Jrz,
(13.37)
allows four different possible combinations for forming
y.
This
method is useful for
determining the number of sheets, but
does
not guide you in determining how they
are
connected.
In
our previous example, with
w
=
(3
-
a)1/2(z
+
2)'12,
the square
roots have only two combinations for
_w,
-
w
=
2fiJrz,
and
so
there were
only
two Riemann sheets associated with
this
function.
(13.38)
13.1.4
Riemann
Sheet
Mapping
Quite often, it is useful to see how
an
entire Riemann sheet gets mapped from one
plane onto another.
This
is quite easy for the function
w
=
z"~.
If
we define
g
=
rzei'
and
=
r,,,ei4,
we can
write
rw
=
Jr,
(13.39)
(13.40)
e
+P=-
2'
According to Figure
13.10,
we
are
on the first Riemann sheet if
0
<
8
<
2rr.
Therefore, the values of
y
we can generate
from
the first sheet will be limited to the
angular range
0
<
4
<
m,
or equivalently the upper half plane. Likewise, the second
Riemann sheet will map onto the lower half plane of
the
w-plane.
This
situation is
shown in Figure
13.25.
Now remember, we said the actual placement
of
the branch cut is arbitrary,
as
long as it connects
the
branch points. For
this
function, the branch cut could equally
well have been placed along the positive imaginary axis of the z-plane. In that case,
the z-plane sheets would map onto the _w-plane as shown in Figure
13.26.
Now the
first Riemann sheet is associated with the range
7r/2
<
8
<
5rr/2,
and the second
sheet with
5~/2
<
8
<
9m/2.
In
the
w
plane,
the
dividing line has been rotated by
r/4.
Keep in mind, there is nothing that says the branch cut has to be a straight line.
It
could be placed as shown in Figure
13.27.
In
this
case, it
is
impossible to define
the different Riemann sheets of the g-plane using
a
simple range of
8.
Don't make
the common mistake of assuming
a
Riemann sheet is always defined by an angular
range.
It is possible that, in order
to
get a one-to-one mapping, there must be multiple
sheets in both the
z-
and
y-
planes. Consider the function
y
=
z3l2.
We investigated
this
function briefly at the beginning of
this
chapter. We showed there were two
values of
y
for a single value of
z,
-
so
there must
be
two :-plane Riemann sheets.
MULTIVALUED FUNCTIONS
533
w-plane
Multisheeted ?-plane
Figure
13.25
Riemann Sheet Mapping for
=
gl/*
-
w
-
Plane
Multisheeted
z
-
Plane
Figure
13.26
Another Riemann Sheet Mapping for
=
g1l2
534
ADVANCED
TOPICS
IN
COMPLEX
ANALYSIS
Top sheet
of
the?-plane
Figure
13.27
Arbitrary-Shaped
Branch
Cut
Quite similar to the square root function, tours in the g-plane reveal two branch points,
one at
z
-
=
0
and one at
=
00.
In
this
case, however, the inversion of
this
function
z
=
-
w2I3
is also multivalued. Each value of
w
produces three different values
of
z,
so
there are three Riemann sheets in the pplane.
Tours
in
the g-plane show that there
are two branch points, one at
y
=
0
and one at
=
m.
If
the cuts in both planes are
taken along the positive, real axes, the _z-plane sheets map onto the y-plane sheets, as
shown
in
Figure 13.28. The branch cut in the z-plane maps onto the real axis
of
the
second
Riemann
sheet
of
the y-plane.
13.1.5
Integrals
of
Multivalued
Functions
Remember, the primary reason we have spent all
this
effort defining branch points,
branch cuts, and Riemann sheets
is
so
we can use the Residue theorem and other
Cauchy techniques to
perform
integrals on multivalued complex functions.
Consider a contour integral
in
the complex g-plane of
our
favorite multivalued
function:
1
dgg1’2.
c
(13.41)
From
our
previous experience with
this
function, we know there
are
two Riemann
sheets,
and
two branch points in the g-plane, one at
5
=
0
and the other at
z
=
a.
For purposes of
this
discussion, let
C
extend across the lower half of the first sheet
of the z-plane,
so
that the contour looks as shown in Figure 13.29. The integrand in
Equation
13.41
is analytic everywhere except at
z
=
0,
and
so
the contour
C
can be
deformed without changing the value of the integral,
as
long as
it
is not moved across
z
=
0.
Let’s attempt to move
it
up, into the upper half plane.
As
the contour is moved
up it is snagged by the nonanalytic point at
z
=
0.
But that
is
not all that happens.
The left half of the contour, along which the real part of
z
is less than zero, moves up
with no problem. But
as
the right half of
this
contour is moved across the positive real
MULTIVALUED FUNCTIONS
535
Multisheeted
-
z-plane
Figure
13.28
Mapping
of
Riemann Sheets
for
=
z3I2
Y
Sheet
#I
Y
Sheet
#2
X
Figure
13.29
An Integration in
a
Multisheeted z-Plane