Chapter 4
Integration
Everybody knows that mathematics is about miracles, only mathematicians have a name for
them: theorems.
Roger Howe
4.1 Deﬁnition and Basic Properties
At ﬁrst sight, complex integration is not really anything diﬀerent from real integration. For a
continuous complexvalued function
φ
: [
a,b
]
⊂
R
→
C
, we deﬁne
Z
b
a
φ
(
t
)
dt
=
Z
b
a
Re
φ
(
t
)
dt
+
i
Z
b
a
Im
φ
(
t
)
dt.
(4.1)
For a function which takes complex numbers as arguments, we integrate over a curve
γ
(instead
of a real interval). Suppose this curve is parametrized by
γ
(
t
)
, a
≤
t
≤
b
. If one meditates about
the substitution rule for real integrals, the following deﬁnition, which is based on (4.1) should come
as no surprise.
Deﬁnition 4.1.
Suppose
γ
is a smooth curve parametrized by
γ
(
t
)
, a
≤
t
≤
b
, and
f
is a complex
function which is continuous on
γ
. Then we deﬁne the
integral of
f
on
γ
as
Z
γ
f
=
Z
γ
f
(
z
)
dz
=
Z
b
a
f
(
γ
(
t
))
γ
0
(
t
)
dt.
This deﬁnition can be naturally extended to
piecewise smooth
curves, that is, those curves
γ
whose parametrization
γ
(
t
),
a
≤
t
≤
b
, is only piecewise diﬀerentiable, say
γ
(
t
) is diﬀerentiable on
the intervals [
a,c
1
]
,
[
c
1
,c
2
]
,...,
[
c
n

1
,c
n
]
,
[
c
n
,b
]. In this case we simply deﬁne
Z
γ
f
=
Z
c
1
a
f
(
γ
(
t
))
γ
0
(
t
)
dt
+
Z
c
2
c
1
f
(
γ
(
t
))
γ
0
(
t
)
dt
+
···
+
Z
b
c
n
f
(
γ
(
t
))
γ
0
(
t
)
dt.
In what follows, we’ll usually state our results for smooth curves, bearing in mind that practically
all can be extended to piecewise smooth curves.
Example 4.2.
As our ﬁrst example of the application of this deﬁnition we will compute the integral
of the function
f
(
z
) =
z
2
=
(
x
2

y
2
)

i
(2
xy
) over several curves from the point
z
= 0 to the point
z
= 1 +
i
.
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