Let $f\in L^1_{loc}(\mathbb{R})$ a locally integrable function, with $f:\mathbb{R}\rightarrow\mathbb{C}$. We say that $\int_0^{\infty} f(t) dt$ converges if the limit
\begin{equation} \lim_{b\to\infty}\int_0^b f(t) dt \end{equation} exists. Now, consider the function
\begin{equation} F(p)=\int_0^{\infty} f(t)e^{-pt}dt \end{equation} with $p\in\mathbb{C}$ such that $p=\sigma +i\tau$. We define the convergence abscissa as
\begin{equation} \sigma_c=\inf_{F(p) \text{converges}} \sigma \end{equation}
Analogously, we define the absolute convergence abscissa, $\sigma_{ac}$. With that said, I'm trying to find a a function $f$ in the conditions established before such that $\sigma_c <\sigma_{ac}$, but I don't came up with any idea.
Could you guys told me a function that verifies this condition or a reference where I can find it? I'd appreciate your help!
Thank you very much!