Graphics Reference
In-Depth Information
Consider the “function”
12
.
wz
=
(10.65)
This w is actually not a function of z because it is not single-valued. Over the
reals
one would normally break it into two functions
()
=+
()
=-
[
)
wz
z and wz
z z
,
Œ
0
,
•
,
+
-
with their domains restricted to the nonnegative reals. Here ÷ is the usual real-valued
function defined on nonnegative reals that returns the nonnegative square root of its
argument. If one allows complex values, a function version of w for negative real z is
()
=+
(
]
wz
i z z
-
,
Œ -•
,
0
.
0
It is easy to see that the two functions w
+
(z) and w
0
(z) would then define a nice
continuous function defined on all of the reals. Things get more complicated if one
considers z as a complex number. In that case, writing z in the polar form
i
q
z
=
r e
,
with
r
,
q
Œ
R
and
0
£
r
,
we can express two single-valued functions or “branches” for w in the form
()
=
re
i
q
2
wz
,
(10.66a)
1
()
=-
re
i
q
2
wz
.
(10.66b)
2
The problem we run into now is one of continuity, or lack of it, because in the
complex plane there are many paths from one point to another. For example, one can
start at e
0
= 1 and then get back to that same point by walking along the points
e
iq
, 0 £q£2p, of the unit circle in a counter clockwise fashion. The function w
1
(z)
starts out as +1 but approaches -1 at the end. What this basically shows is that it is
not possible to define a continuous single-valued square root function on the unit
circle.
Returning to the issue of single-valuedness, one could, of course, pick one or the
other of the branches and forget about the other one, but this would be an unsatis-
factory solution since the multiple-valuedness of the function is an important aspect
of it. Therefore, with the domain of functions restricted to the complex plane, one is
stuck with a concept of multiple-valued function. On the other hand, we shall see that
if one allows enlarging the domain of a function to a “Riemann surface,” then one
can restore the single-valuedness of functions. Historically, searching for a continu-
ous way to pass from one value of a function at a point to another first lead to the
concept of analytic element and analytic continuation, something introduced by K.
Weierstrass.
Let us call a pair (f,c) an
analytic element with center c
if f is an analytic function
in a neighborhood of c. We know f has a power series expansion
2
()
=+
(
)
+
(
)
fz
a
a z c
-
a z c
-
+
...
(10.67)
01
2
