#
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
. |