Graphics Reference
In-Depth Information
1
2
11
24
◊
◊
11 3
246
◊◊
◊◊
2
3
()
=+
(
)
-
(
)
(
)
wz
1
z
-
1
z
-
1
+
z
-
1
- ...
1
and since w
2
(z) =-w
1
(z), it has a different series expansion.
Given a global analytic function F = {(f,c)}, associate to each equivalence class of
(f,c) the well-defined value f(c). This defines a well-defined single-valued function on
the
Riemann surface
, which is the set of analytic branches of F. The Riemann surface
has a natural topology associated to it and is in general a legitimate surface. One can
visualize the Riemann surface in the case of the square root function in (10.65) as
follows (see Figure 10.14): We superimpose two copies, called sheets, of the complex
plane on top of each other and cut each along the positive x-axis starting at the origin.
Think of the edge of the cut along the upper half plane side as the upper cut and the
other edge as the lower cut on each branch. Then starting at point
A
on the lower
sheet (the branch defined by w
1
(z)) we continue around to the points
B
,
C
,
D
,
E
, and
F
on the same sheet. Leaving
F
we get to the lower cut of the first sheet. There we
jump to the upper cut of the second sheet (defined by the function w
2
(z)) and con-
tinue on to
G
on the second sheet. Then move on to
H
,
I
, and
J
on the second sheet.
From
J
we reach the lower cut of the second sheet. At that point we jump back to the
upper cut of the first sheet and arrive back at
A
. The positive axis we cut along is
called a
branch line
and the origin is called a
branch point
. Actually, we could have
cut along any curve that starts at 0 and goes to infinity.
Because we are dealing with multiple-valued functions, rather than expressing
them in the functional notation w = f(z), it is more convenient to think of them and
their associated Riemann surface as defined implicitly by the equation
-
()
= 0.
wfz
(10.69)
The Riemann surface is actually a complex manifold, meaning that its coordinate
neighborhoods are defined by analytic functions (rather than just differentiable real-
valued functions). Basically, this means that every point on the surface has a neigh-
borhood where we can parameterize both variables z and w by a single (complex)
parameter t in the form
sheet 2
H
C
D
I
0
G
B
J
F
A
E
sheet 1
The Riemann surface for z
1/2
.
x
Figure 10.14.
