Complex Integration with Cauchy Formula

Question

This exercise is taken from E. Freitag, Complex Analysis.

Compute, using the Cauchy integral theorem and the Cauchy integral formula, the following integrals: \begin{align*} \int_{\gamma} \frac{\mathrm{e}^{-z}}{(z+2)^{3}}\mathrm{d}z \end{align*} with $\gamma(t)= 3 \mathrm{e}^{2\pi i t }$ $t\in [0,1]$. I calculated \begin{align*} \int_{\gamma} \frac{\mathrm{e}^{-z}}{(z+2)^{3}}\mathrm{d}z = 2 \pi i \mathrm{e}^{2} \end{align*} but in the solutions they calculated $\pi i \mathrm{e}^{2} $ ? I don't find the mistake.

Answer 1

Take $f(z)=e^{-z}$. Then using the Cauchy integral formula $$\int_\gamma\frac{e^{-z}}{(z+2)^3}dz=\int_\gamma\frac{f(z)}{(z-(-2))^{2+1}}dz=\frac{2\pi i}{2!}f^{(2)}(-2)=\pi if^{(2)}(-2).$$ Now note that $f^{(2)}(z)=e^{-z}$.