Cauchy's integral formula for derivatives
Theorem 2.2 (Cauchy’s integral formula for derivatives). Under the hypotheses of Theorem 2.1, \(f\) has derivatives of every order in \(D\), each of them analytic in \(D\), and for every \(z_0\) inside \(\Gamma \)
\[ f^{(n)}(z_0) = \frac {n!}{2\pi i} \oint _\Gamma \frac {f(z)}{(z - z_0)^{n+1}} \dd z \qquad (n = 0, 1, 2, \dots ). \]
Source: [1, ch. 4]