Find the Taylor series for a complex function
Solution 1:
You have$$(\forall n\in\Bbb N):\frac1n=\int_\gamma\frac{f^{(n)}(z)}z\,\mathrm dz=2\pi if^{(n)}(0),$$and therefore$$\left(\forall n\in\Bbb N\right):\frac{f^{(n)}(0)}{n!}=-\frac i{2\pi n.n!}.$$So, the Taylor series of $f$ centered at $0$ is$$-\sum_{n=1}^\infty\frac{iz^n}{2\pi n.n!}.$$