Chapter 4
Integration
Everybody knows that mathematics is about miracles, only mathematicians have a name for
them: theorems.
Roger Howe
4.1
Defnition and Basic Properties
At frst sight, complex integration is not really anything diFerent ±rom real integration. ²or a
continuous complex-valued ±unction
φ
:[
a, b
]
⊂
R
→
C
,wedefne
°
b
a
φ
(
t
)
dt
=
°
b
a
Re
φ
(
t
)
dt
+
i
°
b
a
Im
φ
(
t
)
dt .
(4.1)
²or a ±unction which takes complex numbers as arguments, we integrate over a curve
γ
(instead
o± a real interval). Suppose this curve is parametrized by
γ
(
t
)
,a
≤
t
≤
b
. I± one meditates about
the substitution rule ±or real integrals, the ±ollowing defnition, which is based on (4.1) should come
as no surprise.
Defnition 4.1.
Suppose
γ
is a smooth curve parametrized by
γ
(
t
)
,a
≤
t
≤
b
, and
f
is a complex
±unction which is continuous on
γ
.Thenw
edefneth
e
integral of
f
on
γ
as
°
γ
f
=
°
γ
f
(
z
)
dz
=
°
b
a
f
(
γ
(
t
))
γ
°
(
t
)
dt .
This defnition can be naturally extended to
piecewise smooth
curves, that is, those curves
γ
whose parametrization
γ
(
t
),
a
≤
t
≤
b
, is only piecewise diFerentiable, say
γ
(
t
) is diFerentiable on
the intervals [
a, c
1
]
,
[
c
1
,c
2
]
,...,
[
c
n
−
1
,c
n
]
,
[
c
n
,b
]. In this case we simply defne
°
γ
f
=
°
c
1
a
f
(
γ
(
t
))
γ
°
(
t
)
dt
+
°
c
2
c
1
f
(
γ
(
t
))
γ
°
(
t
)
dt
+
···
+
°
b
c
n
f
(
γ
(
t
))
γ
°
(
t
)
dt .
In what ±ollows, we’ll usually state our results ±or smooth curves, bearing in mind that practically
all can be extended to piecewise smooth curves.
Example 4.2.
As our frst example o± the application o± this defnition we will compute the integral
o± the ±unction
f
(
z
)=
z
2
=
±
x
2
−
y
2
²
−
i
(2
xy
) over several curves ±rom the point
z
= 0 to the point
z
=1+
i