 Solution

The more general version weights each residue by the contour's winding number, with the usual requirement that the contour be homologous to zero in the domain of analyticity containing it and the nonsingular region.

For the requested application, assume

f

is holomorphic on a neighbourhood of the contour and its closed interior, and

z_{0}

is strictly inside it. Its Taylor series shows

\frac{f ( z )}{( z - z _{0} ) ^{2}} = \frac{f ( z _{0} )}{( z - z _{0} ) ^{2}} + \frac{f ^{'} ( z _{0} )}{z - z _{0}} + O (1) .

(2)

Only the second term contributes a residue. The residue theorem therefore proves the derivative form of the Cauchy integral formula:

\frac{1}{2 πi} \oint_{C} \frac{f ( z )}{( z - z _{0} ) ^{2}} d z = f^{'} (z_{0}) .

(3)

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