Convergence of oscillatory integral to real number

Question

Given the integral $$\frac{1}{2\pi}\int_{-\pi}^\pi\mathrm{d}t\,\mathrm{e}^{-s(\omega(t)+\mathrm{i}\sigma(t))}$$ with $\omega(t)\geq0$ and $\omega(t)\in\mathcal{C}^\infty(\mathbb{R})$, is it possible to choose a $\sigma(t)$ such that it converges to a real value larger than zero for $s\to\infty$? I.e. such that $$\lim_{s\to\infty}\frac{1}{2\pi}\int_{-\pi}^\pi\mathrm{d}t\,\mathrm{e}^{-s(\omega(t)+\mathrm{i}\sigma(t))}=\text{const.}>0$$ I'm particularly interested in the case when $\omega$ is a polynomial of an even order.