More Examples in the Use of Cauchy's Integral Formula to Evaluate Integrals
Cauchy's integral formula states that
whereis analytic onifis inside the contour
whereis analytic onifis outside the contour
whereWe can rearrange this formula ro give- with the conditions above - and use it to evaluate any closed contour integral of the above form.
Example: Findwhereis the contour
is inside the contourso
Example: Findwhereis the contour
is outside the contourso
A slightly more complicated example is provided by the case where the denominator factorises.
Example: Findwhereis the contour
We can write
is analytic onandso
Alternatively we may use partial fractions to rewrite the integral.
The first integral is evaluated:
The second integral is evaluatedsinceis outside the contour
Thenas before.