Cauchy's integral formula states that
where
is analytic on
if
is inside the contour
whereis analytic onifis outside the contour
where
We can rearrange this formula ro give
- with the conditions above - and use it to evaluate any closed contour integral of the above form.
Example: Find
where
is the contour
is inside the contourso
Example: Find
whereis the contour
is outside the contourso
A slightly more complicated example is provided by the case where the denominator factorises.
Example: Find
whereis the contour
We can write
is analytic on
and
so
Alternatively we may use partial fractions to rewrite the integral.
The first integral is evaluated:
The second integral is evaluated
sinceis outside the contour
Then
as before.