I am studying semigroup theory and in the proof of Hille-Yosida theorem, we have the integral $$\int_{\omega'-i\infty}^{\omega'+i\infty} \frac{e^{\lambda t}}{\lambda^3}R(\lambda;A)A^3u \, d\lambda$$ Where $A$ is a linear operator, $R(\lambda;A)$ its resolvent operator and $\omega'>\omega$ (we know that the spectre of $A$ is contained in the half space {$\Re(\lambda) \leq \omega$}).
My question is why this integral vanishes for $t\leq 0$ ? I think we can use Cauchy's integral theorem but i don't know how.