Graphics Reference
In-Depth Information
2
p
2
p
(
)
2
p
n
i
w =
e
=
cos
+
i
sin
.
n
n
E.3
Complex Integration
The easiest way to define integration of complex-valued functions is to reduce the
problem to integration of real-valued functions.
Let f : [a,b] Æ
C
be a function. Writing
()
=
()
+
()
ft
ut
i
vt
,
define
b
b
b
Ú
()
Ú
()
Ú
()
f t dt
=
u t dt
+
i
v t dt
.
a
a
a
More generally, let g : [a,b] Æ
C
(=
R
2
) be a C
1
parametric curve and let f(
z
) be a complex
function that is continuous on G =g([a,b]).
Definition.
Define the
line integral of f along
g, Ú
g
f, by
b
Ú
Ú
Ú
()
(
()
)
¢
()
f
=
f
zz
d
=
f
gg
t
t t
.
g
g
a
Using the change of variables formula for integrals it is easy to show that the
line integral is invariant under changes of parameter. What this means is the
following:
E.3.1. Theorem.
Let f : [c,d] Æ [a,b] be a one-to-one C
1
map with f¢ > 0. If l : [c,d]
Æ G is the reparameterization of g(t) defined by l(t) =g(f(t)), then
Ú
Ú
f
=
f
.
l
g
Proof.
See [Ahlf66].
A fundamental theorem of complex analysis is
E.3.2. Theorem.
(The Cauchy Integral Formula) Let f(
z
) be a function analytic on
an open disk
D
. Let g : [a,b] Æ
D
be a closed proper C
1
parametric curve and let
z
be
a point of
D
not in the image of g. Then
()
-
1
2p
f
z
()
=
Ú
f
z
d
z
.
i
z
z
g
Proof.
See [Ahlf66].
E.3.3. Theorem.
With the same hypotheses as in Theorem E.3.2 one has
