Applications of the residue theorem on integration
First we
deduce Cauchy’s integral formula for simply connected
domain D from the residue theorem:
Let
f(z) be analytic in a simply connected domain
D containing a simple closed curve C with a
inside C. Observe then that the function
is
analytic except at a where the residue is
at
;
that is, f(a). By the residue theorem
or
which is Cauchy’s integral formula.
Example. Next we
consider the example of evaluating
by the residue theorem:
Let
be large and
be small. Now let
represent the curve
Consider the
integral
where
The integrand is analytic in C except for a simple
pole at i with residue
=.
On the negative
x-axis
and so
But
and
so
and setting r=x in the first integral we obtain
Now
as
.
A similar argument with r replaced with R
gives
as r.
Consequently,
letting
and
and equating the real parts gives
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