Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin \mathbb{R}$,
\begin{equation} \int_{-\infty}^{+\infty} \frac{1}{t-z}dt=2\pi i, \end{equation}
\begin{equation} \int_{-\infty}^{+\infty} \frac{f(t)}{t-z}dt=2\pi i f(z) \end{equation} for $f$ holomorphic?
\begin{equation} \int_{-\infty}^{+\infty} \frac{u(t)}{t-z}dt=Re(2\pi i f(z))=-2\pi v(z), \end{equation} for $u$ harmonic and $v$ toe harmonic conjugate of $u$?