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.

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