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
.
42
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CHAPTER 4.
INTEGRATION
43
(a) Let
γ
be the line segment from
z
= 0 to
z
= 1 +
i
.
A parametrization of this curve is
γ
(
t
) =
t
+
it,
0
≤
t
≤
1. We have
γ
0
(
t
) = 1 +
i
and
f
(
γ
(
t
)) = (
t

it
)
2
, and hence
Z
γ
f
=
Z
1
0
(
t

it
)
2
(1 +
i
)
dt
= (1 +
i
)
Z
1
0
t
2

2
it
2

t
2
dt
=

2
i
(1 +
i
)
/
3 =
2
3
(1

i
)
.
(b) Let
γ
be the arc of the parabola
y
=
x
2
from
z
= 0 to
z
= 1 +
i
. A parametrization of this
curve is
γ
(
t
) =
t
+
it
2
,
0
≤
t
≤
1. Now we have
γ
0
(
t
) = 1 + 2
it
and
f
(
γ
(
t
)) =
t
2

(
t
2
)
2

i
2
t
·
t
2
=
t
2

t
4

2
it
3
,
whence
Z
γ
f
=
Z
1
0
(
t
2

t
4

2
it
3
)
(1 + 2
it
)
dt
=
Z
1
0
t
2
+ 3
t
4

2
it
5
dt
=
1
3
+ 3
1
5

2
i
1
6
=
14
15

i
3
.
(c) Let
γ
be the union of the two line segments
γ
1
from
z
= 0 to
z
= 1 and
γ
2
from
z
= 1 to
z
= 1 +
i
. Parameterizations are
γ
1
(
t
) =
t,
0
≤
t
≤
1 and
γ
2
(
t
) = 1 +
it,
0
≤
t
≤
1. Hence
Z
γ
f
=
Z
γ
1
f
+
Z
γ
2
f
=
Z
1
0
t
2
·
1
dt
+
Z
1
0
(1

it
)
2
i dt
=
1
3
+
i
Z
1
0
1

2
it

t
2
dt
=
1
3
+
i
1

2
i
1
2

1
3
=
4
3
+
2
3
i .
The complex integral has some standard properties, most of which follow from their real siblings
in a straightforward way. To state some of its properties, we first define the useful concept of the
length of a curve.
Definition 4.3.
The
length
of a smooth curve
γ
is
length(
γ
) :=
Z
b
a
γ
0
(
t
)
dt
for any parametrization
γ
(
t
),
a
≤
t
≤
b
, of
γ
.
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 Math, dt, Manifold, Mathematical theorems, Jordan Curve Theorem

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