Let \(\gamma\) denote the curve \(|z - 1| = 2\), oriented counterclockwise. Evaluate \[ \int_{\gamma} \frac{e^z\,dz}{z^3}. \]
Solution
The value is \(\boxed{\pi i}\).
1. Cauchy’s derivative formula applies at zero.
The circle bounds \(B=\{|z-1|<2\}\), and zero belongs to \(B\) because \(|0-1|=1<2\). The function \(F(z)=e^z\) is holomorphic on a neighborhood of the closed disk. For the specified counterclockwise orientation, Cauchy’s formula for the second derivative gives \[ F''(0)=\frac{2!}{2\pi i}\int_\gamma\frac{F(z)}{(z-0)^3}\,dz \] [@SS03]. Since \(F''(0)=e^0=1\), rearranging gives \(\int_\gamma e^z/z^3\,dz=(2\pi i)/2=\pi i\).