Recovering an analytic derivative from the boundary real part ($f^{\prime }(0)=(\pi r{)}^{-1}{\int }_0^{2\pi }\mathrm{Re}f(r{e}^{i\theta })e^{-i\theta }d\theta$)

For a holomorphic function on a neighbourhood of a closed disc, parametrize the Cauchy derivative formula on its boundary. The conjugate Cauchy integral theorem gives $\int \overline{f(r{e}^{i\theta })}{e}^{-i\theta }d\theta =0$. Replacing $f$ in the derivative integral by $f+\bar{f}=2\mathrm{Re}f$ therefore gives the displayed identity. The boundary real part determines every nonconstant analytic coefficient; only an imaginary additive constant is lost.