We discuss next what appears to be a simpler notion
of integral over a curve.
In this one, we really do regard the curve CC as a subset of the complex plane as opposed to two-dimensional real space;
we will be integrating complex-valued functions; and we explicitly think of the
parameterizations of the curve as complex-valued functions on an interval [a,b].[a,b].
Also, in this definition, a curve CC from z1z1 to z2z2 will be distinguished
from its reverse, i.e., the same set CC thought of as a curve from z2z2 to z1.z1.
- Definition 1:
Let CC be a piecewise smooth curve from z1z1 to z2z2 in the plane C,C,
parameterized by a (complex-valued) function φ:[a,b]→C.φ:[a,b]→C.
If ff is a continuous, complex-valued function on C,C,
The contour integral of f from z1z1 to z2z2 along CC will be denoted by
∫Cf(ζ)dζ∫Cf(ζ)dζ or more precisely
by ∫Cz1z2f(ζ)dζ,∫Cz1z2f(ζ)dζ,
and is defindd by
∫
C
z
1
z
2
f
(
ζ
)
d
ζ
=
∫
a
b
f
(
φ
(
t
)
)
φ
'
(
t
)
d
t
.
∫
C
z
1
z
2
f
(
ζ
)
d
ζ
=
∫
a
b
f
(
φ
(
t
)
)
φ
'
(
t
)
d
t
.
(1)
REMARK
There is, as usual, the question about whether this definition depends on the parameterization.
Again, it does not.
See the next exercise.
The definition of a contour integral looks very like a
change of variables formula for integrals.
See (Reference) and part (e) of (Reference).
This is an example of how mathematicians often use
a true formula from one context to make a new definition in another context.
Notice that the only difference between the computation of a contour integral and an
integral with respect to arc length on the curve is the absence of the absolute value bars around the factor φ'(t).φ'(t).
This will make contour integrals more subtle than integrals with
respect to arc length, just as conditionally convergent infinite series
are more subtle than absolutely convergent ones.
Note also that there is no question about the integrability of f(φ(t))φ'(t),f(φ(t))φ'(t), because
of (Reference).
ff is bounded, φ'φ' is improperly-integrable on (a,b),(a,b), and therefore so is their product.
- State and prove the “independence of parameterization”
result for contour integrals.
- Prove that
∫Cz1z2f(ζ)dζ=-∫Cz2z1f(ζ)dζ.∫Cz1z2f(ζ)dζ=-∫Cz2z1f(ζ)dζ.
(2) - Establish the following relation between the absolute value of a contour integral and
a corresponding integral with respect to arc length.
|∫Cf(ζ)dζ|≤∫C|f(s)|ds.|∫Cf(ζ)dζ|≤∫C|f(s)|ds.
(3)
Not all the usual properties hold for contour integrals,
e.g., like those in (Reference) above.
The functions here, and the values of their contour integrals, are
complex numbers, so all the properties of integrals having to do with positivity and inequalities,
except for the one in part (c) of Exercise 1, no longer make any sense.
However, we do have the following results for contour integrals,
the verification of which is just as it was for (Reference).
Let CC be a piecewise smooth curve of finite length joining z1z1 to z2.z2.
Then the contour integrals of continuous functions on CC have the following properties.
- If ff and gg are any two continuous functions on C,C, and aa and bb are any two complex numbers, then
∫C(af(ζ)+bg(ζ))dζ=a∫Cf(ζ)dζ+b∫Cg(ζ)dζ.∫C(af(ζ)+bg(ζ))dζ=a∫Cf(ζ)dζ+b∫Cg(ζ)dζ.
(4) - If ff is the uniform limit on CC of a sequence {fn}{fn} of continuous functions,
then ∫Cf(ζ)dζ=lim∫Cfn(ζ)dζ.∫Cf(ζ)dζ=lim∫Cfn(ζ)dζ.
- Let {un}{un} be a sequence of continuous functions on C,C,
and suppose that for each nn there is a number
mn,mn, for which |un(z)|≤mn|un(z)|≤mn for all z∈C,z∈C,
and such that the infinite series ∑mn∑mn converges.
Then the infinite series ∑un∑un converges uniformly to a continuous function,
and ∫C∑un(ζ)dζ=∑∫Cun(ζ)dζ.∫C∑un(ζ)dζ=∑∫Cun(ζ)dζ.
In the next exercise, we give some important contour integrals,
which will be referred to several times in the sequel.
Make sure you understand them.
Let cc be a point in the complex plane, and let rr be a positive number.
Let CC be the curve parameterized by φ:[-π,π-ϵ]:Cφ:[-π,π-ϵ]:C defined by φ(t)=c+reit=c+rcos(t)+irsin(t).φ(t)=c+reit=c+rcos(t)+irsin(t).
For each integer n∈Z,n∈Z, define fn(z)=(z-c)n.fn(z)=(z-c)n.
- What two points z1z1 and z2z2 does CC join,
and what happens to z2z2 as ϵϵ approaches 0?
- Compute ∫Cfn(ζ)dζ∫Cfn(ζ)dζ
for all integers n,n, positive and negative.
- What happens to the integrals computed in part (b) when ϵϵ approaches 0?
- Set ϵ=π,ϵ=π, and compute
∫Cfn(ζ)dζ∫Cfn(ζ)dζ for all integers n.n.
- Again, set ϵ=π.ϵ=π. Evaluate
∫Ccos(ζ-c)ζ-cdζand∫Csin(ζ-c)ζ-cdζ.∫Ccos(ζ-c)ζ-cdζand∫Csin(ζ-c)ζ-cdζ.
(5)
