How to Integrate with Cauchy's Derivative Formula

This video shows how to use Cauchys Derivative Formula .
Complex Analysis here where z=x+iy .
Its a neat way of integrating without actually doing any integration .
First see that the integral is set in a Simple Closed Contour which can be set in a simply connected contour . We set C={z:|z|=1} and R=Complex Plane .
the numerator has cos (z-pi) and z^3 in the denominator.
Then we find the simple poles of the function being integrated , and see if they are in the regions specified . If they are outside of the closed contour then they amount to 0 by Cauchys Integral Formula . The ones inside are left which become of interest .
Then we calculate the function as z approaches zero which is the same as eliminating the singularity we are interested in , and multiply by 2i pi .
The second derivative of cos(z-pi) is the only tricky part , and not forgetting the k=n-1 for the factorial value .

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