5.1 Contour Integration
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Fix a domain and continuous. We are going to define
where is any smooth path. Such an integral will be called a contour integral.
Definition (Path Integral)
Given a smooth path the path integral of along is
where the right-hand side is a Riemann integral. This means that the integral is formally defined as a limit
of Riemann sums. As such, we can calculate these integrals using the rules of Calculus, treating like any other constant.
Example
Define by and by . Calculate the contour integral of along .
Solution:
We first compute
and
so that
using the usual rules of integration.
Definition (Contour Integral)
Fix a domain and continuous. Given a contour in we define the contour integral of along to be
which is permitted because each of the is a smooth path.
Contour integration in linear in the sense that both relationships
are true for any continuous and any .
It many not be clear why we should define path integrals the way we have. The definition we have given turns out to be correct for various reasons. First of all, we show how our definition is independent of our speed along and only depends on our route.
Definition (Reparameterization)
Fix a smooth path . A reparameterization of is any smooth, increasing bijection .
If is a reparameterization of then defined by is a smooth path satisfying and . Thus and begin and end in the same places. Moreover, they trace out the same route. They only differ in how quickly or slowly they traverse the route.
Example
Define by . It is a smooth path from to travelling uniformly around the top half of the unit circle. Define by . It is a reparameterization of because it is a smooth, increasing bijection of the unit interval. The path has the formula and also travels from to by travelling around the top half of the unit circle. However, it does not do so uniformly: it is much slower in the beginning (near ) and much faster at the end (near ) than is.
Theorem
Fix a domain and continuous. Let be a smooth path. Then
for every reparameterization of .
Proof:
We calculate
and then apply the substitution to get
as desired.
The real and imaginary parts of our contour integral have interesting physical meanings. Writing
and we can calculate
and try to understand its real and imaginary parts. Write . Let be the vector field
on . We see that
is the work done against in moving along and that
is the flux of across . Here is the unit normal to defined by .
The vector field associated to is called the Polya vector field of .