Graphics Reference
In-Depth Information
E.2
Analytic Functions
Definition.
Let f(
z
) be a complex function defined in a neighborhood of a point
z
0
.
The
derivative
of f(
z
) at
z
0
, denoted by f¢(
z
0
), is defined by
(
)
-
()
f
zh z
h
+
f
0
0
¢
()
=
f
z
lim
0
h0
Æ
assuming that the limit exists.
Definition.
A function f(
z
) defined on an open set
U
is said to be
analytic
or
holo-
morphic
on
U
if its derivative f¢(
z
) exists at every point of
U
. A function f(
z
) defined
on an arbitrary set
A
is said to be
analytic
or
holomorphic
on
A
if it is analytic on an
open set containing
A
.
E.2.1. Example.
Let n be a positive integer. The function
n
zz
()
=
f
is analytic on
C
and
¢
()
=
zz
1
.
n
-
f
n
Although the definition for the complex derivative looks just like that of the deriv-
ative of real functions, it is much more constrained. Let
f
()
=+
z
f x
(
i
y
)
=
(
x y
,
)
+
i
v x y
(
,
)
.
(E.1)
If we approach
z
first along a line parallel to the x-axis and next by a line parallel to
the y-axis, it is easy to show that
∂
∂
u
x
∂
∂
v
x
∂
∂
u
y
∂
∂
v
y
¢
()
=+ =- +
f
z
i
i
.
From this we get the Cauchy-Riemann equations
∂
∂
u
x
∂
∂
v
y
∂
∂
u
y
∂
∂
v
x
=
and
= -
.
(E.2)
Therefore, that u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations is a
necessary condition for f(
z
) to be differentiable. The converse is essentially also
true.
E.2.2. Theorem.
If the functions u(x,y) and v(x,y) of an arbitrary function f(
z
)
expressed in form (E.1) have continuous partial derivatives in a neighborhood of a
point
z
0
, then f(
z
) has a derivative at
z
0
if and only if the Cauchy-Riemann equations
(E.2) hold at
z
0
.