## Using Cauchy's nth Derivative Formula

Letbe a simply connected region, letbe a simple closed contour inand letbe a function analytic onThen for any pointinsideis n – times differentiable atand for n=1,2,3,...

We can use Cauchy's nth derivative formula to evaluate integrals by rearranging into the form

Example: Evaluatewhereis the contour

We have to differentiatetwice and evaluate the result at

Then

Example: Evaluatewhereis the contour

We have to write the integral in terms of its partial fractions first.

Solving the last two equations forandgives

The integral becomes