I know that the following analog to the Cauchy integral formula holds.
For $z$ in the upper half-plane, and $ h_z(\zeta) = \frac{1}{2 \pi i} \left( \frac{1}{\zeta - z} - \frac{1}{\zeta - \overline{z}}\right) $ we have $ f(z) = \int_{-\infty}^{\infty} f(t)h_z(t) dt $
To prove this, evaluate the integral via residue calculus. I think another way to prove this is to start with the Cauchy integral formula for a circle and use a conformal mapping between the unit disc and the upper half-plane to get the above formula.
I would think a similar analog of the Cauchy integral formula for derivatives holds.
Note the upper half-plane excludes the real line.
I know this does not address your exact question, but I hope it is helpful in some way.